Online poker - rigged or not? A case study: Pokerstars by Ionut Apahideanu

2017

ONLINE POKER - RIGGED OR NOT? A CASE STUDY: POKERSTARS

IONUÈ&#x161; APAHIDEANU, PhD 0

The story My relationship to poker has been a sinuous one for as far as I remember. In hindsight, it seems like that kind of first love that one, unable, or unwilling, to forget, keeps returning to repeatedly over a lifetime. I remember first trying the game back when I was a kid, maybe eleven-twelve years old. I was playing with my cousin - the classic five cards draw. Back then, it was the only type of poker we knew. Nowadays there are so many versions of the game, one can hardly keep track of them. I also vividly remember how, in order to make every hand more spectacular, more adrenalin-laden, like the ones we saw in the movies, we used to eliminate from the deck all cards lower than seven. Thus, we got to enjoy plenty of "sensational" duels Aces full of Jacks vs. three Queens, Kings quads vs. the nut flush, etc. Nowadays, they seem to be called "coolers" and omnipresent on online poker sites. But let's not go there, yet. Time passed, with me occasionally playing the game with friends, until I got to university. There, and then, I rediscovered poker. Or vice versa, can't say exactly. We engaged in the most passionate relationship possible. True love, what can I say! It lasted for about two years. Then time passed again, with me getting older, marrying, getting a job - you know, the usual stuff. Until the dawn of the Golden Age of poker: Chris Moneymaker winning the WSOP main event. All of a sudden, everywhere I looked around, it seemed the whole planet was playing Hold'em! Specialized sports' TV stations were broadcasting tournaments or high stakes cash games, there was talk everywhere about some hand Ivey had won against Antonius, Negreanu was becoming the most popular player around, the "Poker Brat" was having a problem with Northern European players to the amusement of TV viewers worldwide, and it seemed that everyday a new poker platform popped up on the Internet. It was simply impossible for me to resist the temptation. I downloaded the Pokerstars platform, the first one I found on the Internet, and I started playing again. Only this time solely on play-money: I felt I had gotten older, or shall we say 'matured', and didn't need the adrenalin of real money anymore. In parallel, I met with friends on poker nights. We had all bought one or another version of a poker kit, and played whenever we could, as a hobby. It was fun! Gradually, the popularity of the game decayed worldwide. One day, I remember replacing my desktop computer with a new one, when noticing the poker application on the old one. I asked myself: do I still need this? Nah, barely played it the last two years, and most likely won't need it anymore. Almost another decade passed yet again, with me playing the game on maybe a handful of occasions, live, among friends, before I once again returned to poker, but this time in more serious manner. Feeling a bit rusty after the prolonged break, I re-installed the Pokerstars application and granted myself about two months of playing on virtual, playmoney, to get back in shape, before jumping to real-money games. Eight to ten, sometimes twelve or more hours a day. Simultaneously, every day I kept (re-) studying the game theoretically, devouring every book or article I could get my hands on: the psychology of the game, video analysis, the tactics, the mathematics. Admittedly, the first time during this training period that I lost three times in a row holding pocket Kings, it felt a bit awkward, and even more so after I calculated the probability of this chain of "accidents" happening. But, still, I had barely restarted playing, right, and accidents do indeed happen in nature, right? The next day or so, at a six-players table, three players had been dealt hole cards of the same suit, all of them eventually making a flush. It did make me raise an eyebrow, but, again, this accident was also "possible", as 1

some online poker spokesperson always say. Another day, in another session, I bumped into a higher pocket pair three of the five times. Again, I jumped into calculating the probability. The next day, I lost with Aces against Kings, on the river. A few minutes later, however, my pocket Queens defeated the opposition Aces. And so on, for the whole two months or so of training. Things seemed a little bit dubious, but all my theoretical and practical knowledge of probabilities, my very background of a researcher made me 'know better': the sample of hands played was still arguably small, randomness in nature implies even such "accidents", overall the figures could have been normalized, etc. Convincing enough for me to dismiss all of the online allegations that were claiming online poker in general and Pokerstars in particular would be "rigged", allegations that I had begun finding on the Internet, I didn't find any solid, scientific and therefore credible demonstration of such preposterous-seeming claims. Moreover, despite those awkward events that I had occasionally witnessed on the platform, I thought my training was going better than I hoped for: I had more than doubled my initial stack of play-money, gaining almost 20k big blinds in a couple of months. All these elements considered, I threw in a couple hundred dollars (not more, given my remnant suspicions), then also bought online a poker data-analysis software, and started playing against the "grinders" at the lowest stakes levels. With no effort of readapting to everything a real money game supposes, things were going smoothly. True, already on my first day of playing (April the 9th), I flopped directly an Aces full, then I lost with pocket Queens against KJ off-suite, the opponent flopping a trip of Kings, but only eight minutes later, I was once again dealt pocket Queens, this time winning the pot, etc. In retrospect, maybe it was the fact that I had ended the session on a little profit that made me not pay much attention to some otherwise suspect events. Things went on and after a good ten thousand hands played, I was on profit, and in full line with my elaborated plan of gradually moving up through the stakes in order to cash in enough to pay the entry fees at live tournaments. All the way to the WSOP series! However, despite everything going according to plan, I couldn't stop noticing that I had already witnessed some truly horrendous "accidents" in terms of anything that a genuinely "random" generator of numbers would actually suppose. Some of them were so truly grotesque, that I instantly screen-shot them, posting of few on my social media accounts. I had for instance already been through Aces vs. Kings duels twice as often than normal at a 6players table. I had seen some of the most improbable flops imaginable, the kind of one in tens of millions of hands. Within pair vs. pair duels I seemed to win visibly more often as an underdog than expected to. And my pre-flop all-ins were completely off the charts, the majority of them having an outcome contrary to the mathematically normal and expected one. I went back to what still seemed like nothing more than online hearsay. I revisited all the hundreds of allegations and complaints already read and explored other thousands of new ones. On the basis of my mere 10k hands played, plus other over 60k played on play-money, I could relate to many of them. What was being said there had also happened to me! And things on Pokerstars did indeed seem a little "fishy", to say the least. Still, I was torn. My inner voice of a researcher kept telling me that at least some of the events witnessed were hard to classify as anything closely related to randomness, and that the situation definitely needed to be investigated scientifically, while the voice of the poker player in me pushed me to keep playing, to win more money and follow my initial plan. Eventually, the researcher's voice prevailed. I threw in another USD 150 or so, speculating a deposit bonus offered by the platform, then won some profitable Spin & Go-s, gathering enough money to even start experimenting things, without the risk of going bankrupt. Thus, contrary to what any rational poker player would do, I started calling my opponents tens of times, despite all the signals telling me I was already beaten, only to reach showdown, so that I could check if another "accident", like AA vs. KK vs. some third pocket pair, or more generally some other form of "cooler" had once again, absolutely "randomly", happened on Pokerstars. Whenever I was losing too much money on these little experiments at the lowest 2

stake levels, I climbed a level or two, where I went back to playing seriously, meaning winoriented, in order to make again enough money to continue testing things. And the more I played, the more showdowns I got to see, and I discovered new anomalies. In fact, any frequency, any indicator, any parameter I was regularly checking did not seem right: from the frequency of triple pair situations to the one of flopping at least a set when holding a pocket pair; or from the conversion rate of my flopped flush draws into flushes up to the recorded rate of winning as a favorite on the turn calculated as a percentage of the expected rate indicated by my equity. Somewhere along the way, probably half way through my series of 55k hands played on real money, I gave up entirely any intention of making money on the platform. My new goal was clear: produce a serious, solid, statistically-based investigation of how truly "random" Pokerstars' algorithm is dealing the cards. I once again returned to the thousands of complains spread all over the Internet, this time evaluating them in a more applied manner and trying to reformulate them in the form of statistically verifiable hypotheses. Gradually, the design of the research was outlined. Afterwards the methods and items to be tested were expanded and refined. As a sample volume, I targeted a number of 50k hands played. I ended up having played over 55k. As a final step taken during the last three days of playing, I even engaged into a full-scale experiment over more than 2.5 k hands: every time I had been dealt a pocket pair, regardless which one exactly, I simply open-limped, then check-called on each and every street, with the sole purpose of dragging my opponent(s) to showdown, where I could see his/her hole cards, which enabled me to expand and refine the data gathered. In petty terms, let's put it like this: at the end of this enterprise, my net financial loss, all things considered, barely amounted to USD 110. It has been one of the most profitable investments I could have possibly made: it has led to a research that represents, to the best of my knowledge, the first statistical investigation of an online poker platform done by an external and truly independent source and simultaneously made freely available to anyone in the world. Above anything else though, the truth is the truth. One cannot put a price tag on it. And if the game on a poker platform is rigged, then the truth has to be known. All the players still actively engaged in online poker deserve the truth. The game of poker - that kind of first love that one cannot or will not forget, deserves the truth. The following presents this research.

Contents I. INTRODUCTION .................................................................................................................................... 6 I.1 The (more than) methodological challenge ................................................................................... 6 I.2 What and how this research investigates ...................................................................................... 9 I.3 Main hypotheses tested and limits of interpretation .................................................................. 14 I.4 General approach, structure, and nature of findings .................................................................. 18 II. HOLE CARDS' DISTRIBUTION ............................................................................................................. 21 II.1 General statistical analysis of the hole cards' distribution on the platform ............................... 21 II.2 Targeted analysis ......................................................................................................................... 24 II.2.1 Suited hole cards' distribution ............................................................................................. 24 II.2.2 Hierarchical groups of hands in relation to their dealing frequency ................................... 28 II.2.3 Pocket pairs' distribution ..................................................................................................... 29 II.3 Pokerstars' "coolers" ................................................................................................................... 32 II.3.1 Coolers.................................................................................................................................. 32 II.3.2 Pocket pair vs. pocket pair: a Pokerstars "classic" ............................................................... 38 II.4 An experiment: (open-) limp-check-calling with pocket pairs all the way to showdown ........... 50 II.5 Summary of the findings ............................................................................................................. 53 III. THE FLOPS: when you take the bait ................................................................................................. 56 III.1 An overall look at the flops on Pokerstars ................................................................................. 56 III.2 A case study: my pocket Queens ............................................................................................... 65 III.2.1 The data commented .......................................................................................................... 65 III.2.2 Main findings ....................................................................................................................... 72 III.3. A quadruple targeted analysis of Pokerstars' flops ................................................................... 82 III.3.1 Flopping a set or better when starting with pocket pairs ................................................... 83 III.3.2 Flopping a flush (draw) when starting with suited hole cards ............................................ 84 III.3.3 Flopping a straight (draw) when starting with middling connectors .................................. 86 III.3.4 Flopping at least one pair when holding un-paired hole cards ........................................... 88 III.3.5 A retesting of the leveling the field hypothesis .................................................................. 90 III.4 Summary of findings .................................................................................................................. 96 IV. AFTER THE FLOP: when things go south ........................................................................................ 101 IV.1 An analysis of six variables on the post-flop streets ................................................................ 104 IV.1.1 Open-ended / double inside straight draws turned into straights ................................... 106 IV.1.2 Flopped inside or semi-open straight draws turned into straights .................................. 108 IV.1.3 Flopped sets turned into full houses or quads ................................................................. 109 IV.1.4 Flopped two pairs turned into full houses ........................................................................ 110 IV.1.5 Flopped flush draws turned into flushes .......................................................................... 110 IV.1.6 Backdoor flush draws turned into flushes ........................................................................ 111 IV.2 Integrating the findings and re-contextualization ................................................................... 112 4

IV.3 The "Riverstars" (sub-)hypothesis............................................................................................ 122 V. A DEEPENED (RE-)TESTING OF THE LEVELING-THE-FIELD HYPOTHESIS .......................................... 130 V.1 A summary of the findings so far .............................................................................................. 130 V.2. Pair vs. pair duels ..................................................................................................................... 132 V.3 A methodological problem and a revisiting of my limp-check-call experiment ....................... 135 V.3.1 A methodological mini-case study: my pocket KK hands .................................................. 136 V.3.2 A revisiting of my limp-check-call experiment ................................................................... 143 V.4 Pre-flop heads-up all-ins ........................................................................................................... 145 V.5 Pre-flop equities in relation to hands' outcomes over a sample of 2k showdowns ................. 151 V.5.1 My perspective................................................................................................................... 155 V.5.2 The favorites' perspective .................................................................................................. 156 V.5.3 A parenthesis: tournaments .............................................................................................. 161 V.6 Integrating the findings ............................................................................................................. 167 VI. CONCLUSIONS ................................................................................................................................ 171 APPENDIX 1 ......................................................................................................................................... 179 APPENDIX 2 ......................................................................................................................................... 180

I. INTRODUCTION "Online poker - rigged or not?" Quite an interrogation, right? Most crucially: how does one actually establish whether it is rigged or not? And, on second thought: what would "rigged" mean, precisely? It is exactly the nature and scope of such a line of questioning and its implicit methodological issues raised that may explain why, from this very beginning and all the way through its four chapters up to the Conclusions, this undertaking may look like something totally unusual. It is not, strictly speaking, an academic paper, since it does not fully comply with the rigors of academic writing1. But nor is it an investigative, non-academic, report, since all concepts, methods and instruments employed are entirely and directly transferred from the academic, scientific world. Let's put it like this for the moment: it's none of these entirely, but it's simultaneously both of them. It is at least something for sure: a pioneering effort. Despite thousands, if not tens of thousands, of complaints spread all over the Internet about online poker in general, and the Pokerstars platform as a case in particular, being "rigged", so far, to the best of my knowledge, there has been no single attempt, by a truly independent source2, at providing a rigorous, extensive, and ideally credible demonstration of one or more of the accusations formulated in social media discussions. As a consequence, all that remained essentially within the realm of public debate has been something like a dialogue of the deaf: demoralized players, thousands of them deserting Pokerstars in recent years, kept accusing the platform of being rigged, while the latter's representatives kept delivering the same, already standardized, excuses as answers, such as: online, people play far more games than in real life, so that those terrible "accidents" invoked by so many critics would in fact be only natural to occur over so many hands; people tend to remember only, or to a higher rate at least, those situations when they were unlucky, and not also the ones where they have been lucky; the samples of hands gathered as evidence by one or another criticizing player have been too small to allow any reliable generalizations, etc. Such answers seemed to only further frustrate many critics, with the accusations escalating and expanding slowly, but surely, up to the point where pejorative terms such as "Riverstars" or "Jokerstars" actually became so widespread, that they turned into distinct hashtags on Twitter and even made their way into urban dictionaries 3. To this, Pokerstars retaliated by apparently hiring an entire legion of paid "drones" entrusted with the sole task of countering criticism and "cleaning" the company's reputation all over the Internet.

I.1 The (more than) methodological challenge The conflict continues to this day. No serious and credible statistical testing has been advanced to break the deadlock one way or the other. And it was only after concretely engaging in the enterprise detailed below that I have understood why. No wonder it hasn't been done before: aside from the tremendous amount of necessary work, the actual, coreproblem is one of a (more than) methodological nature: how does one reformulate online accusations, so differently expressed, respectively understood, by people all over the world wide web, in a form that allows a rigorous scientific testing, and would be approved as such by the scientific community, while nevertheless trying permanently to also make it fully understandable for poker players who are not familiarized with statistics? And therein lies the 1

Some expressions in the text may seem frivolous, there are bold-marked fragments in the middle of a text and other "unorthodox" editing instances, partially anecdotic analogies, numerous screen captures of hands played, meant to simplify the understanding of the reader, etc. 2 Meaning any other source than the one company who "audits" the world's biggest online poker platforms. 3 See for instance http://www.urbandictionary.com/define.php?term=RiverStars (retrieved July 16, 2017).

problem: in order to adequately tackle the problem addressed in this investigation, one would normally need two simultaneous sets of knowledge: of poker and of statistics. Reckoning that such a double requirement would inevitably and severely narrow down the audience, whereas the problem of online poker potentially being rigged seems a problem of a much bigger impact, and as such of interest to a considerably broader audience, I was left with a fundamental first challenge, the one of bridging, as much as possible4, a serious gap between two types of potential readers. Thus, one the one hand, readers who know solely statistics, but not poker, may for instance not understand why in some instances, classic methods and associated instruments and terms of statistical hypothesis testing (T-test, p and Îą, "false positive", Z-test, etc.) could not, unfortunately, be applied in this investigation - it has to do with the specific way the game is played, and, more importantly, with how a data-storing software actually stores, processes, and displays all data (more about this below). For the same reason, the usual approaches such as testing a "null hypothesis" also encountered certain limits of their applicability. Or, to give just another example as a preface: this first category of readers, unless they know poker, might not fully understand why for instance an analysis of the series of the river cards (or the first cards on the flop, for that matter) across all my games played otherwise a correct randomness testing solution - cannot be applied, since, when looking how exactly the game is played, specifically what hole cards are usually kept and played by most players, it becomes clear that the distribution of the cards on the river street actually should not be expected to be random. One of the biggest issues in this regard has been the otherwise classic problem of dealing with incomplete series of data, and consequently the nature, limits, and reliability risks of extrapolations. Subsequently, but only in part overlapping, over the course of the research, I came up with methodological instruments that may seem difficult to understand at first glance, such as the matrix combining two plus one dichotomies: a) the one between a.1.) hands played effectively by me (meaning hands where I did remain actively engaged in the game to showdown), and a.2.) hands where I folded somewhere along the way before showdown, but at least two players have continued playing all the way till showdown. The latter allows the Hold'em Manager 2 software5, directly connected to the platform's software (within certain limits, obviously) to display a series of information such as: the known hole cards of both the two players having reached showdown, and mine, together with the equities, thus allowing me to somewhat counterfactually establish whether I would have actually won the hand, had I stayed in the game; b) the one between b.1.) hands with a certain "outcome" (a term employed either with reference to the winner or, and more often so, in relation to my hole cards and targeted combination6) vs. b.2) hands with an uncertain outcome, meaning for instance hands that have ended on the flop, or on the turn, before thus knowing for a fact whether I would have hit my targeted combination or not, respectively if I would have won. The latter category raised a whole new category of problems in terms of how to correctly approach the factual/counterfactual outcome, etc.

Meaning that, even considering all my clarifications and simplifications of explanations and interpretations, the reader will still require a minimal knowledge of, especially, the ABC of poker (the hierarchy of combinations, the positions at the table, etc.). I can only hope I have let this amount to be indeed the minimum possible. 5 See below. 6 E.g. the binary outcome of me making, or not, a flush, starting from a flush draw on the flop.

As such, throughout my research, I had to often innovate, and sometimes even build from scratch, (new) approaches, methods and indicators, such as the above matrix, or the alternation of personal and favorite's perspectives in terms of equities and corresponding ways of accounting the results of the analysis, or the 13-lines of cards scale, etc.7 Aside from these methodological issues, for this first category of readers there is potentially the additional problem of some of the very main, most commonly used, terms used by this paper, many of them possibly unknown to a non-poker player, a problem which I already tried addressing preliminary in the table below, and more extensively, whenever I felt necessary, throughout the four chapters of this investigation. On the other hand, readers who only know poker, without however a certain amount of knowledge and skills in statistics and more generally in methodological issues, might find it difficult to form an opinion on some of the significance of the statistical indicators used throughout the analysis: correlation coefficients, standard deviations, squared Rs, probability values and confidence intervals, sample representativeness, kurtosis and skewness8 of a particular distribution of hole cards or of the conversion rates of a draw depending on the type of the hole cards, average recorded frequencies as percentages of the corresponding expected one, etc. Nor may they know how many of the probabilities discussed throughout the analysis are actually calculated (although in many more complicated, or at least unusual, cases, I did display the exact method of calculation in order to also avoid any potential controversy). It is in consideration of this latter category of audience that, without any intention of offending the former one, I have tried, sometimes maybe in a frivolous manner, to simplify the explanations and interpretations as much as possible without altering the very meaning and significance of the findings, by employing various accessible analogies and metaphors, such as mixing boiling and freezing water in order to get a normal temperature, or playing the roulette in a casino, or flipping a coin, or making a poll, etc. Additionally, the same task of trying to make everything understandable to an audience as broad as possible, combined with the pioneering profile of the research and the peculiarities of the reality investigated, may also explain in part the considerable size of this document: a final table for instance, with data reprocessed in chain, which in itself occupies let's say - only a few lines, however required a considerably larger space in advance for me to explain, in an broadly accessible manner, what, why and how I measured something. And now, before moving on to what, under any other circumstances, an Introduction should actually address, allow me to briefly clarify, to all readers, the most commonly used abbreviations or terms used in an adapted meaning9: 7

And it is also in this sense that the present Introduction is an atypical one: if, as customary, it would outline to the needed depth level the methodological approach employed, the Introduction would risk becoming - and this is no exaggeration - as voluminous as two chapters combined. 8 The very second I typed this term, the Microsoft Word dictionary immediately notified me, by underlining it in red, that it does not recognize the word. This illustrates my above point. 9 In addition to these common terms, sometimes, for reasons of simplicity, I have used slang terms maybe unfamiliar to persons not playing poker, such as "quads" (instead of "four of a kind"), "cooler", or certain expressions such as "to hit a two outer", "to draw dead", "to flop a full house", etc.

"Hold'em Manager 2" software open-ended straight draw gut-shot straight draw double belly buster straight draw (a.k.a. double-inside straight draw) Random Number Generator. According to Pokerstars' website, it is "a system, device or module that creates a sequence of apparently unrelated numbers" R squared = "coefficient of determination", essentially signifying what Rsq proportion of the dependent variable's recorded variation can be pinned on, or is predictable from, the independent variable (Pearson) (linear) correlation coefficient correl. "average"; unless differently specified, means "weighted arithmetic mean" "avg." suited combination(of hole cards) XY-s off-suite combination XY-o hole cards the starting hand (the two cards a player is dealt, as opposed to the "board" or "community" cards) "hand" used alternatively in two meanings: 1. as an individual game in itself (e.g. "I have played 51,989 hands in cash games on the HM2 OESD GSSD 2xSD RNG

platform")

2. as a signifier of the 169 versions of hole cards that can be dealt to a player in the Texas Hold'em game. In this second meaning, the term "hand" does not differentiate among the four possible suits, only indicating whether the hand is suited or off-suite. As such there are 169 "hands" corresponding to 1,326 "combinations" possible; e.g. an AA hand consists of 6 possible combinations, a AK-o hand of 12, an AK-s of 4, etc. "street" a term used not in its standard meaning, but as an adapted denominator of a game's four phases: pre-flop, flop, turn, and river (cf. Wiki:) the scaled commission fee taken by a card-room operating a poker "rake" game "equity" the probability of one player to win or tie the hand (as displayed by both Pokerstars and the HM2 software. Both indicate the combined probability of winning or splitting the pot) "favorite" the player best positioned in terms of equity at a certain moment of the game One last specification: unless differently mentioned, the chapters use the same general 4-color shading system in all tables: light green: recorded deviation of < +5% of the normal, mathematically expected, frequency; dark green: bigger than +5%; pink: smaller than -5%; respectively red: bigger than -5%.

I.2 What and how this research investigates With all of the above being necessarily (pre-) clarified, allow me to now properly "introduce" the research: it analyzes the series of 55,320 Texas Hold'em hands that I played between the 9th of April and the 28th of May, 2017, on real money10, on the Pokerstars platform11, under the username "Apasu76". Of these hands, 51,989 have been played in cash games of a zoom-format, all of them in 6-players rings, at various stake levels. The other 3,331 have been played in tournaments, within the same timeframe, in different competitions (Sit&Go, Spin&Go, and, mostly, proper multi-table tournaments (MTTs)) in various numberof-players-formats, from a minimum of three up to a maximum of nine.

10 11

As opposed to so-called "play money". http://www.pokerstars.com

Given certain problems of standardizing probabilities, many of which differ depending on the number of players at the table, most of the analysis covers only my 51,989 hands played in cash games. Whenever methodologically possible (and always specified as such), the analysis is expanded to cover all hands played in both cash games and tournaments.

In terms of the representativeness of this sample of hands, for the reader who is not familiarized with either statistics, or the game in itself, allow me to put it like this: there are 169 distinct hands, respectively 1,326 distinct combinations that a player can be dealt in a Texas Hold'em games. In relation to my combined number of 55,320 games, this means that, on the average, I have been through each starting hand 327.3 times12, and through each starting combination a number of 41.7 times. As technological support of the research I have used the Hold'em Manager 2 software. Available for purchase online, this software stores a multitude of data, including video storage, pertaining to all hands one plays on a mutually agreed poker platform, such as Pokerstars. In addition to the storing function, it also allows certain processing of the data collected. Specifically, for those not familiar with it, it separately stores data from tournaments and cash games, and displays among other things, in different selectable ways of arranging:  the hole cards of the user,  the time and date of each hand,  his actions on the four streets (e.g. "X" for check in the third column below, "F" for fold, etc.),  the board / community cards,  the stakes,  the user's net win or loss,  the pre-flop actions,  the frequency of dealing of all 169 hands (the lower-right smaller icon overlapped), etc.

Crucially, it also has video footage of the hands played. If a certain hand has not been played all the way till showdown, only the user's cards remain visible when double clicking it:

For more details, see the first table of the next chapter.

In the hand screen shot above, holding pocket 4s, I check-folded on the flop, so that all the other players' hole cards remain unknown. If, however, a hand has reached showdown (regardless whether with the user still present at the table, or the hand continued by (at least) two of his/her opponents), HM2 will display all known hole cards and, for the first streets within the game, pot size, playing parameters of the opponents, each player's equity (the red figure below the hole cards), meaning the probability of him/her winning or splitting the pot at the given moment of the game (bottom left), etc:

This is how a hand looks when screen-shot directly on the Pokerstars platform (with me holding 54-s and having folded pre-flop):

And this is how the same hand is displayed in the HM2 database, captured at showdown, with my cards hidden, since I had already folded pre-flop, meaning my equity evolution in the left bottom icon is also not shown:

With - as far as I remember - no more than a maximum of seven exceptions, all screen captures presented throughout the paper are made from within the hands where I was personally involved in the game, and technically from within the HM2 database and not from the platform itself, for reasons of convenience related to the data that are displayed and can be correspondingly processed in an easy manner.

I.3 Main hypotheses tested and limits of interpretation With these technical support issues now also being clarified, let us move forward, and get to the actual point of this enterprise: to test whether online poker, in this case the Pokerstars platform, is "rigged" or not. As said, over the last years there have been thousands of complaints all over the Internet supporting this idea - discussions on online poker forums, mass media articles, innumerous screen and video captures of millions of hands, even a few lawsuits, etc.13 However, exactly because of this spectacular variety of statements, there has never been a clear, unified, precisely formulated, and universally accepted meaning, let alone demonstration, of what "rigged" actually means. Hereby I propose an operational meaning of the term, defined by opposition to what the Pokerstars, and most online poker websites more generally, claim, i.e. that the software operates on the basis of a "Random number generator" (RNG), which, selfunderstood, deals the cards in a random manner. Conversely, in my reading, what the critics argue is exactly that Pokerstars' RNG distributes cards non-randomly.

See for instance, among so many: https://legitorscam.org/pokerstars-is-a-scam/, https://www.pokertube.com/article/is-pokerstars-rigged, http://www.pokerupdate.com/poker-opinion/1122-ispokerstars-rigged/ (see the comments, especially), http://www.pokerscout.com/AllReviews.aspx?id=1, http://forumserver.twoplustwo.com/25/probability/pokerstars-rigged-447349/, http://forumserver.twoplustwo.com/54/poker-beats-brags-variance/pokerstars-rigged-proof-tba-1559406/, https://www.youtube.com/watch?v=2Ty5NmT-4-4, https://www.youtube.com/watch?v=kuBvAWwRY_o, https://www.youtube.com/watch?v=EHQidtck9IY, https://www.youtube.com/watch?v=dt7cfdaTCbg, https://www.youtube.com/watch?v=vSnfaik92pc, https://www.youtube.com/watch?v=92tDwMyn0pI, http://pokerstars-rigged-scam.blogspot.ro/, https://www.complaintsboard.com/complaints/pokerstars-anaheimcalifornia-c366882.html, https://bonuscodepoker.com/aga-pokerstars-battle, or http://forumserver.twoplustwo.com/153/high-stakes-pl-omaha/massive-bot-ring-pokerstars-party-how-spotthem-1537778/

Now, under any normal circumstances, there actually should be no debate and nothing for me to research, since any, and I repeat, any IT specialist will tell you there simply is no such thing as a (truly, genuinely) "random" number generator. Any software operating with such an algorithm is conceived by a human mind, which makes it by default anything, but random. What players should expect at best is a pseudo-random distribution of cards. In this adapted meaning, what an online poker platform could do is at best a satisfactory enough mimicking, or simulation, of the randomness otherwise existent, and measurable as such, in nature. Depending on the degree to which their distribution of cards comes close to what a natural random distribution would look like, poker sites RNGs might subsequently be evaluated individually and hierarchically classified as more or less close to randomness. In direct relation to this new adapted meaning and understanding, when further exploring the allegations made in the online realm, the all-so-frequent term "rigged" would implicitly define a deliberate, systematical, and (statistically) significant distancing from what randomness would imply. And this, as is probably obvious already, suddenly becomes considerably more difficult to verify and, if the case, demonstrate: whereas certain recorded variations and deviations from randomness pose no difficulty in terms of measurement and evaluation, given the existence of multiple exact, precise, evaluations of the normality (i.e. randomness) of a distribution, universally employed and accepted, a demonstration of intent, of deliberation, of premeditation, is a completely different matter. After all, accidents do indeed happen in nature. Similarly, an online poker RNG may suddenly be affected by an unintentional error, by a glitch, etc. Where does true intent begin, and how does one demonstrate it beyond any reasonable doubt? Obviously, there is no single, universally accepted, answer to such a question. And herein lies the second fundamental challenge for the researcher of such a topic: how to interpret and evaluate whatever deviations from randomness may be found during the investigation in such terms that both the scientific community approves and the ordinary poker players understand. Should, for instance, a measured linear correlation coefficient of minus .81, covering 3k+ showdowns, between a player's recorded equity on a specific street of the game and the factual outcome of the hand, meaning him/her winning, losing, or splitting the pot, be enough? Of course, correlation does not equate causation, but what if such correlations are repeatedly discovered throughout the testing, and, additionally, and crucially, they all work the same way, meaning their effect is always the same, for instance making the underdog win more frequently than mathematically expected to? Or what if, for instance, when analyzing a player's pre-flop engaged heads-up all-ins, it is discovered that not only did he actually win at a rate of little below 3/4 of the expected win rate, but also, that this "deficit" has been recorded in seven of the eight methodologically-separated all-in categories? Does this indicate something like a deliberate and systematic non-random action of the software? Or, for fans of statistics: what does a p value of, say, 0.044 suggest and how should one relate to it when interpreting the results of a measurement? Interrogations such as these converge towards a fundamental idea: the heuristic limits of any methodological approach, and hence the need to take all figures for what they are and nothing more, nor less: statistical indicators, usually operating in a specific context pertaining to a certain confidence interval, a certain level of statistical significance, etc. For both potential categories of audience, here is a quote from one of the best books on poker I have ever read - Barry Greenstein's Ace on the River: An Advanced Poker Guide.14 Amusing as it is, this hypothetical situation does seem to capture the very heuristic nature of the above-discussed problem, thus sparing me a considerable and most likely redundant amount of further explanation:

Last Knight: Fort Collins, CO (2005, p. 154).

Someone shows you a coin with a head and a tail on it. You watch him flip it ten times and is comes up heads. What is the probability that it will come heads on the eleventh flip? A novice gambler would tell you, "Tails is more likely than heads, since things have to even out and tails is due to come up." A math student would tell you, "We can't predict the future from the past. The odds are still even." A professional gambler would say, "There must be something wrong with the coin or the way it is flipped. I wouldn't bet with the guy flipping it, but I'd bet someone else that heads will come up again."

On a more serious note, it is within the frontiers of the above-indicated limits of interpretation and adaptations of some common, but otherwise inoperative terms such as "rigged", that I formulated the main hypotheses of this investigation and subsequently tested them. A careful reading, over weeks, of the main allegations roaming on the Internet has led me to summarize the most frequent and serious ones as follows:  the hole cards on Pokerstars would be dealt in a non-random manner; most of these allegations further targeted a specific type of situation - the so-called "coolers" such as, notorious among the platform's clients, AA vs. KK vs. QQ three-way duels;  the pre-flop engaged all-in duels in a heads-up format (meaning 1 vs. 1) would show spectacularly abnormal deviations from anything mathematically expected;  the flop cards would further multiply the "cooler" situations, way beyond anything normal in terms of frequency, by simultaneously providing the players that remained at the table with promising, attractive combinations; Some critics specifically use the term "second shuffle", one supposedly occurring after the pre-flop round is finished.  after the flop, the turn and especially the river cards (hence the nickname "Riverstars") would also be dealt non-randomly, usually to the direct favor of the underdog, who, it is alleged, would win considerably more often than mathematically correct in probabilistic terms - hence frequent terms such as "leveling the field" or "balancing the odds";  roughly every - and here estimations differ - every fifth, tenth, or twentieth hand would be "manipulated" by a non-random acting software, meaning these hands would register an outcome contrary to the mathematically expected one;  many players at the cash games tables (but not only) would actually be bots, some of them acting to the direct benefit of Pokerstars, and with their tacit consent. The general logic underpinning these allegations in terms of purpose, or motivation, is one and the same: these deviations from statistical normality implying randomness are deliberate and systematic and serve one purpose: the increase of the proportional rake the platform collects and is ultimately making a living of: "coolers" will make players bet, raise, and re-raise higher, which translates into a bigger rake to be collected; enticing flops act towards the same final outcome; a insidious assisting of the underdog prevents recreational players to lose too much money too fast, thus protecting the player base and the platform's source of income, etc. With the last accusation, albeit it an extremely serious one, remaining obviously un-testable, at least from my position, I have rephrased and regrouped all the other ones in forms that will hopefully be approved by both the scientific community as being correctly formulated and verifiable and the poker players as accurately reproducing the content of the accusations. Oppositely to the general-umbrella equivalent of a "null hypothesis", which would state that anything pertaining to cards distribution on Pokerstars is entirely random (or 16

at least that the recorded deviations are not significant statistically), the hypotheses thus reformulated and correspondingly tested in this investigation are these:  Hypothesis H1: the "non-randomness" of the hole cards dealing; Pokerstars' RNG distributes the hole cards in a manner that is statistically significantly distant from what a normal, probabilistic, distribution should look like; This hypothesis is formulated solely in reference to a player's hole cards. o Sub-hypothesis H1*: the "coolers"; the recorded frequency of "cooler" situations is significantly higher than the normal one. This sub-hypothesis is formulated in direct reference to opponents and their hole cards.  Hypothesis H2: the "second shuffle" hypothesis; in similar terms, the flop cards are also being dealt non-randomly, in the sense that the factually recorded frequency of flopping useful combinations (for exactly the players that have remained at the table after pre-flop) are higher, category-by-category, than their mathematically expected, normal, counterparts; this includes any new addition of "coolers" or a development of new ones;  Hypothesis H3: the "leveling-the-field" (or "balancing the odds") hypothesis: throughout the entire game, the underdog is directly and significantly assisted by the non-random distribution of cards, in that he/she ends up winning significantly more often than mathematically expected to. This hypothesis is tested solely across the series of cash games, and not also tournaments15, and assumes two meanings, or "dimensions", plus two sub-hypotheses.  a.) the first dimension is defined in relation to an opponent, in the sense that the "underdog", meaning the lower positioned player in terms of probabilities, wins more often than normal, to the detriment of the "favorite", meaning the player at the table with the mathematically highest probability to win or tie;  b.) the second dimension is formulated in relation to one's hole cards, stating that the better those cards, the less they actually end up winning eventually, respectively proportionally to their strength16. o Sub-hypothesis H3A: the "Riverstars" sub-hypothesis; the non-randomly dealt cards on the river are directly responsible for the underdog winning more frequently than probabilistically expected to; o Corollary H3B: "Every tenth hand is manipulated"; roughly every tenth hand that is effectively played (meaning not folded pre-flop by all but one player) is manipulated, meaning hijacked from its natural, normal, mathematically expectable, outcome. By its very formulation not a testable (sub-) hypothesis, but rather an orientative, indicative, statement, it admits at most circumstantial evidences, in the form of more instances in which the discovered deviations come close to the referential 10%. 15

The same online folklore claims that in tournaments the situation would actually be the opposite, meaning favorites would win more than expected. The underlying business-motivated logic is that, unlike the case of cash games, in tournaments, the only source of income for the platform consists of the entry fees paid by participants. As such, the platform would have no interest whatsoever in prolonging the games by leveling the field, quite the contrary: the sooner a tournament ends, the sooner new ones can be opened, with more entry fees being paid. 16 This "strength" is made operational by employing a classification of the general, by default, probability to win or tie a hand for each of the game's 169 hands - a so-called "power ranking" of the hands. For more details, see the first chapter.

I.4 General approach, structure, and nature of findings The above-operated reformulation of online allegations in the form of testable hypotheses has obvious advantages in terms of the potential knowledge to be gained in a reliable, safe, manner, but it also has direct implications on the very structure of the research, one directly conditioned by the hypothesis' formulation and place of testing. Thus, in relation to the four phases of the game, some phenomena associated with the hypotheses are to be identified and verified on the (chronological) longitudinal, whereas others should manifest on the transversal of the phases, which allows neither of two possible ways of approach and subsequent structuring: on the longitudinal, meaning chronologically corresponding to the four streets of the game; on the transversal of the game's streets.

Given the peculiar nature of the phenomenon investigated and the corresponding way in which the general logical approach conditions any possible structuring, I have opted for a solution that once again may deem this investigation as one somewhat atypical: a mixture of longitudinal and transversal approaches: the first three chapters are delineated chronologically, corresponding roughly to the game's phases (pre-flop, flop, and after the flop), but chapters two and three are each supplemented with a transversal investigation grouping all findings related to the leveling-of-the-field hypothesis and its spin-offs, while the fourth, and last, chapter adopts a nominally transversal approach across the four streets by further expanding and deepening the testing of the leveling-of-the-field hypothesis. In the framework of this general mixed approach, the first chapter addresses Hypothesis H1 and its derived sub-hypothesis. It starts with a general statistical testing of the hole cards' dealing in terms of the normality of the distribution and then continues with a targeted analysis of three specific distributions, or items: of suited hole cards; of hierarchically arranged groups of hands; of the infamous "coolers", with a particular emphasis on pocket pair vs. pair duels. Finally, it addresses a very revealing experiment that I tried on the poker platform: (open-)limp-check-calling each time I had been dealt pocket pairs with a twofold purpose in mind - testing the alleged randomness of the hole cards distribution and 18

alleviating the methodological problem of the unknown represented by hands that ended before showdown. The findings not only confirm most of the online allegations, but also expose a spectacularly elaborated, sophisticated, way for the Pokerstars software to permanently keep the final, overall, average parameters absolutely "normal" at the end of the day, although the specific manner in which this is achieved has nothing to do with randomness, quite the contrary, it is rather "anti-random". It is a series of such findings throughout the investigation that strongly hint, and sometimes seem - no matter how much caution the researcher tries to keep - to bluntly indicate, beyond any reasonable doubt, evidences of intent, deliberation, premeditation. The second chapter covers the flops in search of any traces of the alleged "second shuffle" that some claim would occur at the end of the pre-flop. In a first step, it statistically analyzes flops in a general manner. As a second step, all my hands where I have been dealt pocket Queens are instrumented as a case study of the flops. After the latter's summarizing and contextualization by expanding the same analytical approach over other pocket pairs, the third step is a quadruple targeted analysis of four flop situations: 1.) a set or better when starting with pocket pairs; 2.) a flush or flush draw when starting with suited hole cards; 3.) straight or draw when starting with middling connectors; 4.) at least one pair when starting with non-paired hands. These four variables address both the second shuffle hypothesis more generally, and the renewed or extended "cooler" situations in particular. Multiple, statistically significant, mutually reconfirming and further reinforced by the discoveries within the other chapters, the findings of the second chapter cast a huge shadow of doubt over all claims stubbornly repeated by Pokerstars representatives of its software distributing cards in a "random" manner. Moreover, the one-way direction in which the quasi-totality of the recorded anomalies all manifest themselves in a systematical and synergic manner, i.e. towards an increase of the rake collected by the poker operator, considerably adds substance to the online allegations made by critics and further hints at premeditation, at pure intent, at a deliberate conception of the software to act in a certain direction that does not even try to mimic randomness. The third chapter investigates the distribution of cards on Pokerstars in relation to what happens after the flop, meaning on the turn and on the river, in a both longitudinal and transversal approach. In a first step, it builds on the findings of the previous chapter, further measuring the post-flop recorded conversion or improvement rates of six draws or combinations in comparison to the normal, probabilistic, rates to be mathematically expected in a randomness-governed environment: 1.) of flopped OESD or 2xSDs into straights; 2.) of flopped GSSDs straights; 3.) of flopped sets into full houses or better; 4.) of flopped two pairs into full houses; 5.) of flopped flush draws into flushes; 6.) of flopped backdoor flush draws into flushes. The second step isolates and extensively tests, over no less than 3k showdowns, the "Riverstars" sub-hypothesis, one to be verified on the last street of the game. Whereas, as shall be seen, the sub-hypothesis has been definitely disconfirmed in its strict sense, the collateral findings of the research however are some of the most shocking of the entire underlying investigation. In addition, the chapter's findings not only capture significant multiple differences to the events happening on the flop, in contrasts extremely hard to justify if presuming randomness, but also reconfirm both the non-randomness of Pokerstars' flops (via a different path) and the leveling the field hypothesis. The last chapter's investigation is pursued on the transversal of the game's streets, specifically testing, in a deepened and extended approach the "leveling the field" hypothesis, right after summarizing the previously gathered findings. It targets four distinctively addressed items, by permanently comparing recorded win plus split rates with the 19

mathematically normal values to be expected in any environment supposed to be random: 1.) pure heads-up pair vs. pair duels; 2.) a revisiting of my (open-) limp-check-call experiment by focusing on the favorite vs. underdog comparative evaluation of probabilities and recorded results - supplemented by a methodological discussion on the limits of any extrapolation into hands folded pre-showdown; 3.) my entire series of pre-flop engaged heads-up all ins; 4.) a comparison, extended over a sample of over 2k showdowns reached among my played hands, between the pre-flop equities of the favorites at the tables and the final outcomes of the hands in terms of the winner. The findings not only reconfirm the previous, partial, and disparate preliminary conclusions formulated, but also provide a considerably broader and deepened confirmation of the leveling-of-the-field hypothesis in ways and at values of the various coefficients and indicators employed that are in my opinion simply too many and too spectacular to allow any further talk about "accidents", "variance", or any such similar terms so often used by Pokerstars' representatives. Additionally, although nominally outside the scope of the hypothesis tested in the chapter, tournaments that I played are, as a one-time exception, integrated into the analysis of the last chapter in a comparison to cash games operated from the same perspective of the favorite vs. underdog dichotomy. Even if only in light of the shocking nature of the findings, the reader may deem this exception as being justified.

II. HOLE CARDS' DISTRIBUTION When verifying the claimed randomness of a poker platform such as, in this case, Pokerstars, the first thing, in both chronological and importance terms, to be investigated, is the distribution of the hole cards. This chapter specifically addresses this issue. It starts with a general statistical analysis, and continues with a targeted analysis separately covering three items: suited hole cards; hierarchical groups of hole cards; respectively the pocket pairs' distribution. Separately, due to its significance and the shocking nature of the findings, the third step of the analysis addresses the so-called "coolers" situations occurring on Pokerstars, both generally and particularly in regard to pair vs. pair duels. A fourth step presents the results of an experiment I made in May 2017 on the platform, playing 2,573 hands over two consecutive days in a specific and deliberately conceived and applied manner, while the final part summarizes the findings of the chapter.

II.1 General statistical analysis of the hole cards' distribution on the platform In analyzing the hole cards' distribution on the poker platform, the table below highlights the general situation of the combined 55,320 games that I participated in (both the 51,989 cash games and the 3,331 tournament ones) in the timeframe April 9 - May 28, 2017.17 The results are integrated using the usual suited - off-suited - pairs diagonal matrix, with the suited cards occupying the upper-right triangle in relation to the pairs diagonal, and the off-suite hole cards placed in the opposite triangle. For each of the 169 hands / hole cards, the figure placed above in their cell indicates the hand's absolute frequency (i.e. how many times I was dealt that particular hand), whereas the figure underneath it displays its relative frequency, calculated as a percentage of its normal, expected, probabilistic, frequency. For example: within the total 55,320 hands played, I received pocket Aces a total 240 times, which is 95.9% of the normal, expected frequency (which is 0.4525% of 55,320, meaning I should have received Aces 250,32 times)18. In terms of the shading, the colored are used in this manner: Shadings: recorded frequency as % of the expected frequency = below 89.9% = 90 - 94.9% = 95 - 99.9% = 100 - 104.9%

= 105 - 109.9%

= above 110%

At first glance, as displayed below, everything seems to be in order: all 6 colors seem relatively uniformly dispersed across the 26 columns and lines of the tables, while the aggregated figures on the last column and line are all within a reasonable Âą5% margin of variation in relation to their expected, normal, frequencies. True, no less than 20 of the 169 hands' recorded frequencies exceed that margin, which in itself is reason for some concern, but as said, at a superficial look at the table nothing particularly suspicious catches the attention.

All of the hands, as mentioned in the Introduction, being recorded and stored, including video footage, on the Hold'em Manager 2 software I have been using. 18 There are 6 possible pairs (i.e. distinct combinations) of Aces, comprising 0.4525% of the total 1,326 twocards combinations one can be dealt in Texas Hold'em.

Table: My hole cards distribution over 55,320 hands played:

Anyhow, not at peace with this superficial undertaking, I opted to specifically verify if there is any correlation between a hand's power ranking19 (i.e. their win + split rate in a 6-players game, meaning how strong a hand is, by default) and it's recorded frequency of being dealt to me calculated as a % of its normal, expected frequency20. Because power rank(ing)s differ among them depending on the number of players at the table, and the 3,331 hands I played in tournaments have been played in various formats (from 3 to 10 players at the table), I isolated the 51,989 hands played in cash games, all of them in 6-players rings. Convincingly enough, for these hands, there is absolutely no mathematical relation between how strong a hand is and how often it has been dealt to me, the Pearson correlation coefficient being a mere +0.0079, which suggests independence between the two variables beyond any doubt.

For a detailed hierarchy of such power ranks in a 6-players game, see this excellent website: https://wizardofodds.com/games/texas-hold-em/6-player-game/ (retrieved June 27, 2017), the data being also copied in the Appendix to this document. 20 E.g. my AJs hands: normally, there are 4 AJs combinations among the total 1,326 possible in a 52 cards deck, meaning a relative frequency of 0.302%. Applied to the total 51,989 hands I played in cash games on Pokerstars, this would mean a normal, expected, absolute frequency of 0.00302*51989 = 156.83. In the recorded reality, I have been dealt AJs hands 146 times, which means that the recorded frequency represents 146*100/156.83 = 93.09% of the expected frequency. This percentage has been calculated for all the 169 hands, and then correlated to their power ranking.

Graph: hole cards' power ranking in relation to their dealing frequency: (N = 51,989 hands played in cash games, 6-players rings)

Furthermore, even when reframing the entire 169 hands sample analysis in terms of the normality of the dealing frequency, the recorded data convincingly meet the statistical requirements necessary to be considered a normal, Gaussian, probabilistic distribution: Graph: Dealing frequency of hole cards (169 hands) in terms of normal distribution: (N = 51,989 hands played in cash games, 6-players rings)

Thus, aside from the tiny triple mode peculiarity21 (attributable to the size of the statistical population analyzed - 169 individuals = hands), all other parameters are confined within normal intervals of variation. Thus, the mean and median are practically equal. The skewness 21

There are three groups of hands, each equally comprising six cases, which represent the mode. These clusters of 3 cards are overrepresented in relation to their normal, expected frequency, to degrees of 101.3%, 101.9%, and 109.7%.

is weakly positive, indicating a slight tilt of the distribution curve to the right, one however statistically not significant. The excess kurtosis is positive, which means (what is already visible on the graph) a leptokurtic distribution (an aspect already visible on the graph). Finally, the standard deviation - delineated variation intervals of the distribution also look normal: 69.8% of the cases lie within 1σ distance from the mean (vs. 68.27% normally), 94.7% within 2σ (vs. 95.45% normally), the rest of the cases not exceeding a 3σ distance from the series' mean. So, after two in-depth verified statistical indicators both suggesting everything has been normal on the Pokerstars platform, the case should be closed, right? Well, not really. And this is actually a masterpiece of the poker platform's software: somehow, despite the sometimes grotesque statistical anomalies recorded in various instances, overall, at the end of the day, the general, average values of most of the parameters will be normal. But how exactly this is achieved, represents is a completely different story. To put it as simple as possible: it is like mixing boiling water with ice; the two have extreme and opposite temperatures, but, at the end of the day, after all the mixing is done, the recorded temperature is indeed normal, isn't it? Consequently, if some auditor at the end of the day shows up and verifies only this indicator - i.e. the average general temperature, things will be in order. Without prematurely going into details, here is an example that may help clarify the explanation: in the Hold'em game, in a pair vs. pair heads-up encounter, the higher pair is, mathematically, a roughly 81.5% favorite to win the duel. However, as I shall detail later on in this analysis, on Pokerstars, due to a business-motivated strategy of "leveling the field" or "balancing the odds", the lower pair (and more generally, the underdog) clearly, constantly, and significantly wins more often than mathematically expected to. This works both ways, meaning both for you and for your opponent. Thus, when you bump into an opponent with a higher pair, you will win more than 18.5% of times, but so will your opponent in case it is you holding the higher pair! Well, at the end of the day, if you are curious enough to calculate your average equity for all pair vs. pair or any other type of encounter you have been through, you will find out that it is respected, meaning that, for instance, if you have been a 66.6% average favorite to win, you will have indeed won roughly 2/3 of the duels. However, this has been achieved by mixing in opposite statistical and mutually compensating exceptions, meaning you actually win more often than normal when being the underdog and less often than normal when being the top dog!

II.2 Targeted analysis In relation to the above-captured apparent "normality" of the hole cards distribution on Pokerstars, there are at least four clear examples that I can provide as arguments in support of my above statement: • the distribution of all suited hole cards specifically distinguished by the suite type; • the frequency of hole cards along power-ranked hierarchical groups of hands; • the distribution of pocket pairs; • the frequency of pocket pair duels, part of a phenomenon usually nicknamed "coolers". The following addresses these four points with the last one of them, as already mentioned, addressed separately for reasons previously explained.

II.2.1 Suited hole cards' distribution The following continues the above analysis in terms of the statistical normality of the hole cards' distribution, but this time specifically isolating the suited hands I have been dealt in the 51,989 cash games I went through, and, additionally, differentiating among the four possible suits: clubs (♣ - colored in green in the table below), diamonds (♦ - blue), hearts (♥ - red) and spades (♠ - black). Statistically, any of the 1326 two-cards combinations 24

corresponding to a 52 cards deck has an equal chance of being dealt to a player, one of 1/1326 = 0.0754%. This applies to any pocket pair combination (6 possible for each pocket pair hand), to any off-suite combination (12 possible for each off-suite hand), and to any suited combination (4 possible for each suited hand). For my 51,989 games recorded, this means it is expected for each of the 1,326 combinations to have been dealt to me 0.0754%*51,989 = 39.207 times. Here are the recorded absolute frequencies (abbreviated "F") for the 12,299 suited hands that I have been dealt, differentiated by the specific suit and arranged in a decreasing order for each suit, from the most frequent (52 of clubs) to the most rare (K2 of hearts) combination (abbreviated "C"): Table: suited hole cards' frequency of being dealt to me: (out of 51,989 hands analyzed, cash-games, 6-players rings)

Summarizing the findings, I have been dealt clubs-suited hands 3,039 times, hearts 3,174 times, diamonds 3,075 times, and spades 3,011 times. At first glance, the heartssuited recorded frequency stands out among the four, but the recorded surplus in relation to the expected frequency (of 3058) is of only 3.8%, which is indeed considerable, but not worrisome. Also, as detailed below, one should notice that I have been dealt suited hole cards

more often than expected, but, again, not to a significant degree22. Furthermore, even when dissecting and recomposing the aggregated data in terms of the normality of the distribution, the result is almost too good to be true, meaning the distribution is as normal as one could expect, if not more, actually!

Thus, the excess kurtosis is a staggering -0.002, the skewness, although clearly visible above, is half the threshold value beyond which one may conclude some abnormality, whereas mean, mode and median overlap almost perfectly. In fact, even after employing more advanced statistical methods and techniques, such as CDF or Z-value, the difference between the recorded distribution and a mathematically perfectly normal, Gaussian, random-specific, is, as shown below, almost invisible! 60

recorded

expected 20

1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199 208 217 226 235 244 253 262 271 280 289 298 307

Specifically, 12,299 times as recorded vs. 12,232.7 as expected.

And it is exactly this perfect overlapping that is highly suspicious. It is as if Pokerstars somehow periodically makes sure that the distribution of the hole cards on the overall is constantly Gaussian all the way to the third decimal place! The recorded figures are simply unbelievably... "perfect". Simply amazed by these almost-too-good-to-be-true results obtained, I went back to the table listing the dealing frequency of the suited hands. One thing in particular immediately caught my attention, namely the suited connectors. For the readers not so familiar with the Hold'em game, there are probably two explanations required at this point: firstly, the term "connectors" signify any two consecutive-valued cards, such as QJ, 87, or 32; secondly, among all 78 suited hands, the suited connectors are actually the best ones, meaning they are the hands most likely to win a pot, since they can transform, using the board, or community, cards, in either a flush or a straight, aside from their individual values. With this being clarified, a recalculation of the figures recorded above highlights a slight overrepresentation of the 12 suited connectors among all the hands I have been dealt (3.71% recorded vs. 3.62% expected), but a way more interesting finding is the spectacular, indisputable, correlation between the suited connector's power rankings and the frequency of them being dealt to me as hole cards: a negative 0.762, meaning the better the suited connectors, the less often they have been dealt to me23: Suited hand dealt

Hand's Absolute power frequency ranking of dealing (6-players games)

AK KQ QJ JT T9 98 87 76 65 54 43 32 Correl.:

141 159 137 149 153 155 185 174 162 161 178 175 -0.762

32.1 29.5 27.6 26.3 23.9 21.8 20.4 19.3 18.4 17.8 15.9 14.0

Not coincidentally in this light, it is exactly 5 of the best 6 suited connectors whose recorded frequency is significantly below their expected frequency. Anyhow, the coefficient's value is simply too high to attribute such an anomaly to randomness, suggesting instead human, nonrandom, intervention, in the process of dealing the hole cards. Additionally, when looking closer at the very first table of this paper, the one listing all the 169 hands' frequencies, one can easily notice that, of all the total 26 lines and columns of the matrix, two of the four most underrepresented ones are the first line from top (97.8% of the expected frequency), respectively the first column from the left (98.5% of the expected frequency). These are exactly the line and the column corresponding to the Ace card, which needless to mention, is the best possible card in the game! This observation in 23

Readers not really fond of statistics should now the linear correlation coefficient's scale stretches from -1 (total negative, inverse, relation) to +1 (total positive, direct, relation).

turn determined me to take one step further and target the single-card dealing frequency registered on the platform along my combined cash plus tourney 55,320 hands, whose results, although not part of the present analysis of suited hands, I nevertheless display below: Graph: single cards distribution on the poker platform:

The findings speak for themselves: not only it is exactly the Ace, the best card in the game, that has been dealt to me the least often, but, more generally, the low cards have also been more frequent than the high ones. True, one may legitimately start thinking that the last two exposed anomalies (that is the suited connector's conditioning of frequency by their individual value and the single card distribution) should be themselves sufficient to begin thinking of Pokerstars' RNG as one not really" random". Let us however move on with the analysis and, at least for the moment, try to pretend that it still might be randomness involved, in the sense that the two bizarre results could still have been just an "accident". After all, why couldn't one simply be an unlucky guy, whose most rarely distributed single card happens to be exactly the best one in the game, right?

II.2.2 Hierarchical groups of hands in relation to their dealing frequency By keeping in mind the boiling vs. freezing water metaphor and hence ignoring the already established fact that, overall, no correlation has been identified between hole cards' values and their frequency of being dealt to me, the present analysis focuses, for reasons of necessary standardization, only on the 51,989 cash games I have participated and delineates hierarchical groups of hole cards according to their power ranking in 6-players games24. Specifically, the table below focuses on the best 30 of the total 169 hands that can be dealt in a 6-players Hold'em game:

The reader will allow me to remind that a hand's initial power ranking differs depending on the number of players at the table. Take for instance the AK suited hand: in a 6-players game, it is the fifth best hand in the game, with a 31.1% probability to win or split the pot; in a 10-players game, it is the fourth best (20.7%), whereas in a heads-up game, it is only the eighth (67.0%). Such discrepancies are the reason for me isolating only my 6-players cash games, where all figures are standardized, as the tournaments I have participated in were of various formats, from 3 to 9 players rings.

The figures indicate a slight, but still visible, relation between the top hands and the frequency of them being dealt to me; on the overall, the top 30 hands in the game have been dealt to me more frequently than normal; along the groups the five, the first two of them are also over-represented in terms of dealing them as hole cards; the situation normalizes relatively beneath the cluster of the best 25 hands in the game but, after the best 30 hands are counted, the overall cumulated frequency is still slightly higher than the expected one. This means that the overall lack of correlation I discussed above has been the result of some distribution anomalies (big enough to alter the measurable correlation) registered somewhere among the weaker rated hands.

II.2.3 Pocket pairs' distribution Pocket pairs represent one of the most striking issues noticeable on the Pokerstars platform in at least three regards: their dealing frequency - one clearly, constantly, and consistently higher than normal; their probability of encountering at least another pocket pair at the table - one also clearly above normal, as I shall detail at a later point in this analysis; their conversion rate along the next three streets in the game (flop, turn, river) into sets or better - an aspect that I shall also investigate in-depth at later. For now, let us deal with the first aspect and focus on the overall recorded dealing frequency of pocket pairs along my entire 55,320 series of cash and tournament games played on the poker platform, by starting with two specifications: regardless of the number of players at the table, the probability of being dealt a specific pocket pair is 6/1326 = 0.4525%; in aggregated terms, the probability of being dealt any pocket pair is 78/1326 = 5.88%, meaning roughly once in every 17 hands dealt. The discrepancies between the above-mentioned normal frequencies, to be expected in a probabilistic, random environment, and the ones actually recorded on Pokerstars are, as visible in the graph below, quite remarkable. Thus, in comparison to an expected absolute frequency of 250.3 times, the recorded average frequency of pocket pairs that have been dealt to me is 258. On the overall, I have been dealt pocket pairs 3,354 times within a total 55,320 games played, meaning a relative frequency of 6.06% in comparison to the normal 5.88%, this surplus having manifested itself in both cash and tournament games. Clearly, and, as detailed below, continuously, pocket pairs on the Pokerstars platform are being dealt more frequently than normal.

Pocket pairs distribution (No.) (55,320 hands played, cash games + tournaments) 22s

278

277

99s

264

263

88s

261

44s

259

average

258.0

55s

258

255

normal frequency

250.3

33s

250

77s

66s AA

243

242 240

This recorded anomaly is by no means an "accident". It cannot, at least reasonably, be attributed to randomness, or variance, or whatever term Pokerstars usually employs in trying to justify some of the weird things happening on the platform, for at least two reasons. Firstly, as one can notice above, it is not one, not two, not three, but 9 of the total 13 pocket pairs that are situated above the normal frequency to be expected in a genuinely random environment, meaning the overall figures cannot be attributed to some single case outside of the third standard deviation interval, which would alter the recorded average. 30

Neither can these findings be attributed to some short-term incredible variance that might have happened to me. Specifically, because the frequency of pocket pairs and, partly overlapping, the frequency of pocket pair vs. pair situations both already caught my attention after a week or two of playing on the platform, back when I was doing it only on play-money, the moment I started playing at real money tables, I began monitoring the pairs situation and kept a detailed account of it. Here are the results recorded at three separate moments in time in relation to the 51,989 cash games I played: after the first 15,503 hands played (the lower segment of the columns in the graph below); after 29,313 hands played (the middle segment of each column); respectively after 44,312 hands played (the upper segments):

Frequency of pocket pairs dealt along three consecutive timeframes (after 15,503 games; after 29,313; after 44,312) (N = 51,989 cash games played)

250

200

83 64

76 68.3

67.9

150

65 80 100

66 64

77s

66s

55s

44s

33s

22s

66.7

62.4

71.8

70.1

99s

88s

average normal

Using the same 4-colors shading as usual (dark green for over +5% variation in relation to the expected frequency, light green for less than +5%, pink for less than -5%, and dark red for over -5%), one can easily notice, looking at the two columns placed right in the graph, that, regardless which timeframe is chosen, the average recorded frequency of pocket pairs has constantly been higher than the normal, expected one. This suggests intentionality, premeditation, on the part of the people who conceived and approved Pokerstars' cards distribution algorithm, as well as the impossibility of attributing the findings to randomness. The explanation for this anomaly is quite simple and reconfirmed by the findings throughout this analysis regardless what indicator is taken into account: strong, high powerranked hole cards are dealt more often. For the players involved, this simply makes the game more fun, more spectacular, more attractive, which may subsequently lead to an increase of the player-base, which consequently implies more profit for the poker platform. And in this explanatory light, it should be emphasized that, at least in 6-players games, pocket pairs comprise the best group of hands. Thus, not only are five of the best six hands25 in the game pocket pairs (AA, KK, QQ, JJ, and TT), but, statistically, and essentially, a pocket pair holding player will almost always be the favorite at the table unless the one exception-case in which he encounters a higher pair. Otherwise, even a 44 pocket pair for instance will still be favorite (albeit it, narrowly) against an AK-s! Hence, having a pocket pair will tend to make a player bet. In the elaborate scenario in which there is an opponent who also has a pocket pair (more on this in the next subchapter), there is an increased chance there will be some raising and maybe even re-raising. And, logically, bigger pots mean bigger rake for the poker platform to collect. It is, truly, that simple! And therefore it is also no coincidence that on Pokerstars, at least over the 55,320 games analyzed here, pocket pairs, the best group of hands, are dealt more frequently, followed by suits as the second best group of cards, whereas off-suite cards are being dealt below their normal frequency. Before concluding the analysis here, there is another remark to be made: the rarest pocket pair that I have been dealt in the combined 55k games played has been AA, exactly the best pair and also the best hand possible in the game. And, when engaging into calculations, the results indicate that the overall underrepresentation of the Aces as single card cannot be attributed solely to the lower frequency of the AA pocket pairs. This sheds a different light on the finding above (when I invited the reader to give Pokerstars the benefit of the doubt), namely that the rarest single card distributed to me was the Ace; in any normal circumstances, such a coincidence would be highly suspicious; on Pokerstars, as this study demonstrates, it can no longer be considered just a "coincidence".

II.3 Pokerstars' "coolers" Aside from one individual's hole cards themselves, what has been at least equally suspicious to me on the investigated platform, way before engaging in this investigation, is the interactive distribution of hole cards. This refers to the hole cards that two, sometimes three, players are sometimes dealt simultaneously (well, truth be said, not sometimes, but quite often, actually), despite some extremely low probabilities of such events to happen). The following section firstly treats the Pokerstars "coolers" situations in a general manner, and secondly targets specifically the pair vs. pairs situations.

II.3.1 Coolers The slang term "cooler" signifies a situation in which a very strong hand loses, but not because of the player committing a mistake, but against an even stronger hand. As such, 25

Or 7, depending on the source and the type of ranking, i.e. by win/split rate or by the expected value.

the loss is due to a particular, and usually very improbable, exceptional, way in which the deck cards have been "shuffled". Needless to say it again, the one certain consequence of such a cooling distribution of the hole cards is the increase of the pot and therefore a bigger rake for the poker platform to collect. Obviously, the most common, notorious, and simple, example of "coolers" are the AA vs. KK duels, meaning an encounter of the best two hands in the game. However, in the meaning in which it is used throughout this analysis, the term "cooler" does not refer strictly to double or triple high pair situations. It more broadly encompasses any situation in which, pre-flop and/or on the flop, two or more players each have extremely attractive, albeit equally improbable, hole cards or combinations using the flop cards. The hand screen-captured below, with me holding Q4-o, and folding pre-flop, is a perfect example of such a "cooler":

In the example above, two players each flop an enticing trip, while the third one gets directly a full house. The conditioned probability of such a flop? An amazing 0.013%. What do you think happened on such a flop? Well, quite predictably, a first player 26 bet, the second one raised (53-o), the third one (holding 88) called the raise, then the fourth one, holding the A5-o, went all-in, which the third and fourth player called. This is how, in cash games, such "coolers" ensure that people will bet and raise massively, thus increasing the pot-size and subsequently increasing Pokerstars' collected rake and it's profit. An image being worth a thousand words, allow me, before engaging into a detailed analysis, to first display five screen shots of such coolers recorded on Pokerstars. The first two, as an exception, are from games that I did not participate in, but only observed27, the other from games in which I was directly involved in. The first example can be regarded as a cooler not so much because of the hole cards dealt to the three players, but because of the flop:

Let's be clear as possible: (at least) two of the six players at the table have pocket pairs and, additionally, the third has one card of a rank equal to the JJ pocket pair! Extraordinarily, since in the 46-cards remaining deck there are, at best28, one Jack, two 3s and three Kings left (meaning a total 6 cards), the flop brings one of each of these three values on the table. The probability of such a flop, which insidiously encourages all three players at the table to bet, thus automatically also increasing the collectible rake, is a ridiculous 6/15180 = 0.039%, meaning it'll happen once in every 2564 times that these three players will find themselves in the same starting position, which in itself is already a low probability! The second example, captured below, is one that I find astonishing even by Pokerstars standards: it shows us how one can lose a $2,477 pot on this platform, despite having quads! I think a little math is required here: in a four players game, (at least) two players have been dealt pocket pairs, both of them flop a set, then both of them make the quads on turn, respectively river! 26

The one placed right below the 53-o holding player. He folded along the way, so his hole cards remained unknown. 27 The Pokerstars platform offers its users the "observe" option at all the stakes levels, allowing at least a (carefully chosen) selection of hands to be observed. 28 Meaning we assume that these cards have not been distributed to the other two players who had folded preflop!

Let's again explore the associated probabilities, all calculated from the perspective of the player holding the pocket 4s:  the probability of being dealt a pocket pair other than Aces: 72/1326 = 5.43%  the probability of any of the three opponents having Aces 1-[((C50,2)-6)/(C50,2))^3] = 1.47%  the probability of the flop offering both players a pure set: 2*2*44 (useful flops) / 17296 (total flops possible) = 1.02%  the probability of both of them making quads, one (any) of them on turn, the other on river = (2/45)*(1/44) = 0.1515% Thus, the conditioned probability of the entire chain of events happening is a dumbfounding 5.43% * 1.47% * 1.02% * 0.1515% = 0.00000123347%. Formulated alternatively, it means that something like the above will happen once in every 81,071,840.23 hands played! Or: if playing 81 million hands, an event like the one above could be considered as statistically "normal". Now, of course I don't know for sure, since I don't know the pocket 4s holding player, but I may rhetorically ask: would Pokerstars representatives try to convince us that he/she did indeed play eighty-one-point-zero-seven million hands on the platform?! Since such "cooler" situations are seemingly a deliberate policy of Pokerstars, it is only logical that I personally also could not avoid them when playing on the platform. Here are three examples from the games that I was directly involved in. In the first one selected, a hand screen-captured directly on the poker platform, I had been dealt 54s and was smart enough to fold pre-flop, when the raise - re-raise - re-reraise - re-re-re-raise spiral erupted (with two players eventually going all-in pre-flop), which, admittedly, only made me curious enough to check the fold-but-keep-watching game option available on the platform.

With this happening on May 18, after a good 5 weeks spent on the platform, I have to confess I was already convinced even pre-flop that this was another classic AA vs. KK situation, all so frequent on Pokerstars (see below), the only surprise being the third player, who also held pocket Kings. With the mathematics detailed below, let me just specify the probability of such an event happening, as calculated for any of the two players holding Kings: 0.09% of all those situations in which he/she has been dealt pocket pairs! Whereas in the previous example, with a weak hand such as 54s, I made the easy decision to fold, I cannot say the same about the next displayed hand, where I did remain in the game all the way to showdown, only to discover that three players at the table had hearts hole cards, all of them making a flush!

Without delving into boring mathematical formulas again, let me again specify the probability of such an event happening: 20.52% * 12.16% * 5.39% * 22.15% = 0.0298% of all those hands in which one has already been dealt suited hole cards in a 6-players game, or once in every 3,356 suited hands played! As a third example, one of my luckiest hands on the platform, as captured by the HM2 2 software: 36

Spectacular, sensational, unbelievable? I have frankly run out of labels, or, better said, Pokerstars has consumed all the labels on my stock. Let's again get specific: the conditioned probability, when holding pocket Kings in a 6-players game, of bumping not only into the single higher pair (Aces), but simultaneously into a third pair at the table (any other than AA or KK), and, additionally, flopping the quads, is a staggering 0.00026% of all hands of pocket KK played. What this further means is that something like this will happen once in every... 384,705 KK hands played in a 6-players ring! It is probably redundant to specify, but, of all my 51,989 cash games played, the total number of KKs that have been dealt to me was 259 (see capture below), out of which a mere 139 hands have made it to at least the flop! And still, this ridiculously minuscule probability materialized. Or, from the general perspective, considering the frequency of being dealt Kings, an event like the above will happen every 85,111,726 hands dealt, whereas, in reality, the total number of hands that I played in 6-players cash games has been less than 52k!

The problem is not that such events, that, in order to be plausibly considered statistically "normal", require players to have played hundreds of thousand or even tens of millions of hands, do happen. After all, they are improbable, but, anyhow, possible; the core problem is that all of these "coolers" have one single thing in common, aside from their nanoscopic probabilities: they make sure that players will bet, raise, and re-raise, thus guaranteeing bigger rakes, which is exactly what Pokerstars makes a living of! Or, to put it conversely: no "non-cooler", or say we shall "dull", "boring", nonspectacular, situation, has ever been found in this analysis, nor ever reported by any player on Pokerstars for that matter, to have occurred at a frequency lower than its expected one. Say for instance playing 100 KK hands and not encountering Aces at a 6-players table; or playing 1,000 suited hole cards hands and not encounter a situation where three players simultaneously make a flush; or of having 2,000 flopped sets and not encountering an opposition flopped set; or playing 20,000 hands in a 6-ring and not going through the classic AA vs. KK vs. QQ situation; etc. Such "boring" distributions, with people not betting or even folding, fail to achieve a crucial task: the increase of the rake29: it's that simple.

II.3.2 Pocket pair vs. pocket pair: a Pokerstars "classic" Within the broader category of "cooler situations", which comprises, as selectively captured above, various types of situations, such as triple pairs, triple flushes, flopped sets vs. sets, full house vs. flushes, or quads vs. quads, there is one subcategory that, given the particulars of both the Texas Hold'em game and the mechanics of the Pokerstars software, can and needs to be addressed in-depth: the pocket pair vs. pair dealing situations. Mathematically, in a genuine randomness, probabilistic, context, when holding a certain pocket pair in a 6-players game, considering that within the whole 50 cards left deck (meaning 1225 combinations of two cards) there are 73 paired combinations possible30, and there are 5 opponents at the table, the probability that at least one of them also holds a pocket pair, regardless whether it is equal or non-equal to your own, is 26.45%, as can be easily calculated using the formula [1 - ( ((C50,2)-(2*12+1)) / (C50,2) )^5] * 100 = 26.45% Without going into further mathematical detail, let us now specify six other probabilities, associated with the situation of holding a pocket pair in a 6-players game, which will show the relevance of the below findings: ď&#x192;ź the probability of encountering, somewhere among the 5 opponents, a specific pair is 2.425%. For instance, if one holds KK in a 6-players game, the probability that (at least) one of the 5 opponents holds Aces (any of the 6 AA possible combinations) is 2.4251%, that he holds tens the same 2.4251%, etc. ď&#x192;ź the probability that one of the 5 opponents holds exactly the same pocket pair as you is 0.407%. That means if you for instance have been dealt pocket 9s, there is only a 0.407% probability that one, any, of your 5 opponents also holds a 99 pair; ď&#x192;ź the probability of one of the 5 opponents also holding a pocket pair, but one (any of the 12 pair hands left) different to your one is 26.13%. For instance, if you hold pocket

And more so, since, it should be specified, Pokerstars does not collect rake from hands that end pre-flop, with all but one player folding. This type of hands and situations does simply not serve the platform owners' business interest. 30 Six combinations each for the other twelve pocket pair hands possible (meaning 6 AA possible, 6 KK possible, etc.) + 1 left of the same value (e.g. if you have 22s, there is only one other possible combination of 22 left within the deck.

Deuces, the probability that one of your opponents hold a pocket pair any from AA to 33 included, but not deuces, is 26.13%; ď&#x192;ź the conditioned probability of two of the five opponents also having pocket pairs (regardless which) is 5.754%; ď&#x192;ź the probability of two other opponents also having pocket pairs, but with two of the three pocket pairs at the table being of the same rank, is 0.185% ď&#x192;ź the probability of (at least) three players having pocket pairs, and one of the two opponents having a same rank pair with yours is 0.09%. These figures might put some of the situations occurring on Pokerstars in a quite different explanatory light. Specifically, as players usually bet high, and extremely often go all-in with the highest pocket pairs (Aces, Kings, and to a lesser degree, Queens), such pairs' simultaneity subsequently means bigger pots, and consequently more rake for online poker platforms to collect. This is why, on Pokerstars (but, truth be told, also on other platforms I have tried over the course of time), the frequency of such high pair vs. high pair encounters is way above the normal, probabilistic, frequency.

In my case, of the 51,989 hands played in cash games in a zoom format at 6players tables, I held for instance pocket Aces 232 times (see the HM2 capture displayed above). Applying the mentioned normal frequency, I should have met opposition Kings in 2.425% of the Aces hands I played, which would be 5.63 times. Well, I bumped into Kings AT LEAST 10 times, which equates a percentage of 4.3, which in turn means 1.8 times the normal frequency: 1.) April 17, at 17:14 EET; 2.) April 19, at 14:38; 3.) April 22, at 13:01; 4.) April 22, at 13:0931; 5.) May 2, at 15:50; 6.) May 3, at 14:52; 7.) May 21, at 12:04; 8.) May 23, at 12:38; 9.) May 25, at 16:45; and 10.) May 28, at 16:46.

Yes, that's right, on the same day, only 8 minutes after the previous case!

I cannot stress enough the term "at least" which I have used above: these 10 cases are the ones in which the existence of Kings in one of my opponent's pocket can be proven, 40

since both I and my opponent went to showdown and saw each other's hole cards, hence the above screenshots. These cases are certain. However, after having played a combined 110,000+ hands on Pokerstars on both real and play money, and judging by some of the betting patterns of my opponents in a few of the other cases I held Aces, but we didn't reach showdown, I have extremely strong reasons to suspect there were at least 22 other instances in which my opponent held Kings (or - a slight possibility - Queens). Most likely, they had the wisdom to fold, either pre-flop against my 3- or 4-bet, or on the later streets, when it might have become clear to them that I had Aces.32 This would increase the frequency of AA vs. KK encounters to 5.1%, which is closer to the 2.2 (if not 2.5) higher than normal frequency I generally estimate for pair vs. pair situations in 6-players games on the overall on the Pokerstars platform. And if that wouldn't in itself be enough in regard to Pokerstars RNG's supposed "random" dealing of the cards, it should be noted that two of those hands I held pocket Aces and encountered Kings were consecutive, and placed at a time distance of only 8 (eight) minutes (on May 22, at 13:01 and 13:09), as captured below by the HM2 software:

Well... the conditioned probability of bumping twice in row, when having a pair (AA in this case), into the same other pair (KK in this case) is a minuscule (0.2613*00.2425)*100 = 0.63%, meaning once in every 159 hands of that particular pocket pair that are being played, whereas the total number of my pocket Aces that have reached at least the flop has been only 131. And, as repeatedly specified throughout this research, neither can this anomaly be attributed to other pocket pair hands that I've played, that would supposedly compensate it33; take for instance my pocket 8s hands: an event such as the one above has happened not once, not twice, but three times. Specifically, I bumped consecutively into: Kings (May 27, at 19:04, and then on May 28, at 16:25); Queens (May 19, at 16:34 and then at 16:47, so 13 minutes away); respectively 9s (May 10, at 13:18, and then the very next hand on the same day, at 14:17). As a recurrent conclusion of this analysis, in order for some statistics on Pokerstars to be possibly considered statistically "normal", I should have played

See for instance my Aces hands played on: May 26; May 21, at 15:29; May 17, at 16:08 (called 3-bet, folded on turn); May 7, at 12:40 (called 3-bet, folded on flop); April 19, at 14:24 (same playing line); and especially: May 20, at 12:54 (3-bettor who folded in front of my massive 4-bet, April 18, at 10:38 (called 3-bet, folded on river), and April 15, at 14:38 (called my 4-bet, called on turn, folded on river). 33 Meaning that, in relation to the overall number of pocket pair hands that I have been dealt, the total frequency should potentially be normal.

thousands and thousands of hands more than I actually played, sometimes 25-30 times more hands than I actually did! Returning to the item-example of my pocket Aces, overall, out of the total 232 times they have been dealt to me, I reached showdown in 75 cases - enough to discover pair in my opponents' pockets no less than 34 times (Kings, as mentioned, 10 times, Queens 6 times (also a little higher than the normal, expected frequency)34 - see the screenshots below, 7s and 4s three times each, etc.).

Additionally, I encountered another opponent with pocket Aces twice in 232 hands that I played at least on the flop35, which would place such an event on a 2/232 = 0.86% frequency, more than 2 times higher than the normal one of 0.407% (see below for a more detailed discussion of this type of situation). For all pocket pairs, other than Aces36, that I held, it is impossible to irrefutably prove the real percentage of cases in which one of my opponents held another pocket pair, regardless if higher or lower than mine. The obvious reason for it is that both me and my opponents, being rational beings and playing on real money, obviously did not go every time all the way to showdown independent of how the betting line evolved, of what the board cards 34

Which is also above the normal, expected, frequency of such encounters: May 26, at 18:30; May 26, at 14:21; May 20, at 20:46; May 18, at 11:37; May 6, at 11:49; April 20, at 13:47. 35 I use this aggregated figure (flops both played and folded pre-flop) as reference since it's safe to presume that in none of the 101 hands not played beyond pre-flop could an opponent have folded with pocket Aces. 36 Which I folded just one time, for the specific purpose of testing how the RNG will "react" :)

were, etc. Numerous times, either me or my opponents folded along the way when I held pocket pairs (I personally folded two KK hands before showdown with an Ace among the board cards and my opponents betting aggressively both pre-flop and afterwards). As there are no showdowns that could have been screen-captured, I thus unfortunately cannot provide a solid, irrefutable, accounting. However, glimpses of Pokerstars overinflated percentage of pair vs. pair situations can be brought forward by some disparate cases, such as for instance: - in at least 7 of the 259 times I held pocket Kings, when reaching showdown (i.e. 94 times), I encountered Aces37 - meaning 2.7% of the KK hands played, which is already above the normal frequency of 2.42%; also with Kings in my pocket, I encountered Jacks also at least seven times38, and 9s an additional six (three times actually on the same day (May 26), within little over 4 hours of game-play!)39 - of the 249 hands of pocket Queens, 95 did not reach the flop; of the remaining ones, I reached showdown with 86 of them, enough to discover in my opponent's pockets Aces at least 6 times40, same for Kings41, respectively Jacks42; - having pocket Jacks (a total 235 times), I only reached showdown 77 times, enough yet to certify in my opponents pockets: Aces at least 5 times, Kings at least 6 times, 10s at least 5 times, 4s at least 6 times (which beat me 3 times) - a total staggering 37 times that at least one of my opponents also had a pocket pair, of, I repeat, only those cases in which the game reached showdown; - of the 249 hands of pocket 8s I was dealt, 90 of them reached showdown - in at least 36 of them, at least one another opponent also had a pocket pair; thus, I bumped into Tens 6 times, into Queens43 also 6 times (in two of those instances there was also (at least) a third opponent also with a pocket pair!), into Kings 7 times44, etc. Surely, or at least very likely, Pokerstars' representatives might try counterarguing by saying that in the hands that did not reach the flop, or have been played, but folded before showdown, my opponents did not have pocket pairs, so that, overall, meaning hands ended pre-flop plus hands played further, the overall frequencies are actually normal. However, cases such as those when I held AA or KK and did go all the way to showdown any hand I had received, cases in which the recorded frequencies of pair vs. pair encounters are significantly above the normal, expected one, cast a huge doubt over any such possible refuting tentative. These are the specific cases one has to focus on when trying to estimate the frequency of pair vs. pair encounters, since in these cases the percentage of hands taken all the way to showdown is the biggest, hence the images captured by them at showdowns will be the closest to the actual reality on the poker platform. For the lower pairs I held, the % of hands that reached showdown decreases visibly, and subsequently directly alters the reliability of the findings at showdown: 37

May 26, at 18:13; May 25, at 17:18; May 21, at 16:32; May 16, at 15:39; May 18, at 16:09; April 17, at 11:36; April 11, at 15:40. I also encountered Aces once within 18 KK hands that I have been dealt in tournaments (of which 11 made it to showdown). 38 May 26, at 18:13; May 14, at 16:54; May 14, at 15:20; May 10, at 14:54; April 29, at 10:12; April 13, at 17:27; and April 11, at 17:42. 39 I even managed to lose 50% of the times against 9s, despite being a 81.5% favorite. My KK hands when I encountered 9s were played on: May 26, at 17:23, at 12:27, and at 11:14; on May 19, at 13:20; on May 13, at 14:30; and on May 7, at 12:29. 40 May 26, at 15:10; May 25, at 13:31; May 23, at 13:14; May 20, at 12:27; May 19, at 19:10; and April 18, at 00:15. 41 May 22, at 16:31; May 19, at 16:55; May 14, at 16:20; April 21, at 13:58; April 14, at 17:11; and April 11, at 12:30. 42 May 22, at 12:33; May 16, at 12:31; May 16, at 11:45; May 11, at 11:19; April 13, at 15:19; and April 12, at 11:47. 43 May 19, at 16:47; May 19, at 16:34 (so less than 15 minutes distance from the previous encounter); May 13, at 15:52; May 10, at 13:18; May 2, at 11:31; and April 18, at 18:15. 44 May 28, at 16:25; May 27, at 19:04; May 20, at 19:53; May 2, at 11:31; April 26, at 11:22; April 20, at 12:53; and April 16, at 12:32.

My pocket pair

total hands dealt

Aces Kings Queens Jacks Tens 99s 88s 77s 66s 55s 44s 33s 22s total

232 259 249 235 240 252 249 230 222 243 239 237 259 3146

Showdowns reached No. 75 94 87 77 73 88 90 73 66 59 63 59 71 975

% of hands 32.3 36.3 34.9 32.8 30.4 34.9 36.1 31.7 29.7 24.3 26.4 24.9 27.4 31.0

The figures and the associated graph capture a crystal clear tendency: the higher the pocket pair I held, the bigger the chance of reaching showdown with it. This is only logical: despite occasional experiments such as the one separately discussed in this paper, when I open-limped and check-called each time I had a pocket pair, with the sole purpose of dragging my opponent to showdown, in order to test some statistics, in the overwhelming cases of all 51,989 hands that were dealt to me, I did, or at least tried to, play rationally, meaning the stronger the hole cards, the higher my likelihood of staying in the game till showdown. Hence, as argued above, the best available indicators when trying to grasp the frequency of pair vs. pair encounters on the Pokerstars platform are those hands where I held strong pocket pairs, whose percentage of showdowns reached is considerably higher. This logical reframing of the analysis sheds some additional explanatory light on the findings in the next table:

My pocket pair

Showdowns reached

Aces Kings Queens Jacks Tens 99s 88s 77s 66s 55s 44s 33s 22s total

75 94 87 77 73 88 90 73 66 59 63 59 71 975

Showdowns where 1+ opponent had pocket pair No % 34 45.3 38 40.4 33 37.9 36 46.8 25 34.2 29 33.0 34 37.8 25 34.2 16 24.2 25 42.4 17 27.0 16 27.1 23 32.4 351 36.0

The tendency is as clear as possible: the higher my own pocket pair, the higher the percentage of cases with at least one of my opponents also having a pocket pair. True enough, the overall percentage of 36.0% of showdowns highlighting at least two pocket pairs at the table is higher than the normal 26.45% one, but, this general discrepancy could still be attributed to those hands folded before showdown, claiming that in those cases opponents would not have had pocket pairs, so that the overall frequencies would be eventually right. However, when isolating, for reasons explained above, the cases in which I held a strong pocket pair, say the best five pocket pairs (AA to TT), one discovers that in no least of 166 of the 406 showdowns reached, meaning 40.9%!, there was at least one other pocket pair at the table. That's 1.55 times more frequent than normal, in a random environment. And this is based only on those hands in which, simultaneously, both myself and my pocket-pair holding opponent have chosen to stay in the game all the way to showdown! If, however, Pokerstars' representatives would push the speculation, in trying to explain such anomalies, by claiming that the hands folded before showdown must have been compensating the overall statistics, I think the further analyses below shall deem any such tentative as futile.

ď&#x192;&#x2DC; Equal pair encounters' frequency If a general and superficial look at the overall frequency of pair vs. pair duels casts a substantial doubt on Pokerstars' claim of its RNG being random, but still cannot definitively prove non-randomness, other items verified shall definitely clarify the situation such as, for instance, the very frequency of equal pairs being dealt at a 6-players table on the Pokerstars platform (situations as the one screen-captured below), which is also significantly above the normal, expected one of 0.407%

Specifically, as partially highlighted in the table below, I encountered an equal rank pair twice when I held Aces, once for each of the KK, QQ, and JJ, and 44 series of hands I had been dealt, and again 2 times each for my TT and 77 series of hands. In an effort to approximate the frequency of this type of situation on the platform investigated, I isolated the total 1,215 hands of AA, KK, QQ, JJ, and TT that I have been dealt, excluding from the analysis other lower pocket pairs, since in the latter type of case the number of showdowns reached covers only a fraction of the total hands dealt (the weaker the pocket pair, naturally, the lower the chance of me reaching showdown). In turn, the higher pairs cases I selected ensure approximately one third of all hands having reached showdown45. My pocket pairs:

Total hands dealt

Ended preflop

Played, but ended before showdown

Showdowns

Aces Kings Queens Jacks Tens Total

232 259 249 235 240 1215

101 120 95 82 79 477

56 45 67 76 88 332

75 94 87 77 73 406

Equal pair encounters % of hands No. dealt 2 0.86 1 0.39 1 0.40 1 0.43 2 0.83 7 0.58

Equal pair situations displayed at showdown, when I held: Aces - May 19, at 17:51, and May 22, at 11:22; Kings - May 24, at 15:48; Queens - April 14, at 14:52; Jacks - May 26, at 16:50; Tens - May 6, at 17:11, and April 27, at 11:17.

The simple correlation coefficient between my pocket pairs values and their associated % of hands that reached showdowns is +0.772 .

Thus, in 0.58% of the total pocket pairs hands selected that I was dealt, there has been an opponent with an equal pair, a figure already, and significantly so, above the normal, expected one of 0.407%. And, it needs to be stressed, this is by basing the inventory exclusively on the 406/1215 = roughly one third of the hands dealt that did reach showdown. Of the total showdowns reached, no less than 1.7% displayed an equal pair vs. pair situation, which would be over four times the normal frequency! One can safely presume that, among the 2/3 hands that failed to get to showdown, there have been other instances when an opponent (or me, for that matter) also had an equal pair in his/her pocket but, considering how the board evolved, folded before showdown! So what would be the real percentage of equal pairs encounters? Only Pokerstars' employees can answer that. From the outside, even the frequency recorded (solely) at showdowns, but based on all the hands dealt, is already above the normal, expected one! ď&#x192;&#x2DC; Triple pair situations' frequency Aside from the above anomaly, there is another pre-flop recognizable problem with Pokerstars' claim of its algorithm being random: the frequency of triple pairs situations, such as the one captured here:

As mentioned above: the conditioned probability of two of the five opponents in a 6-players game also having pocket pairs (regardless which) is 5.754%; the probability of two other opponents also having pocket pairs, but, specifically, with two of the three pocket pairs at the table being of the same rank, is 0.185%, and the probability of (at least) three players having pocket pairs, and one of the two opponents having a same rank pair with yours is 0.09%. As an example, I will pick the pocket 99s that I have been dealt over the entire 51,989 hands that I played on the Pokerstars platform. Of the total 252 cases of pocket 9s, 68 have ended pre-flop, 96 have been played at least to the flop, but were folded before showdown, while the number of hands taken all the way to showdowns was a mere 88. Well, these 88 were enough to discover a triple pairs situation in no less 4 times (meaning 4.5%!): May 14, at 14:32 (Hand No.1 in the table below); May 13, at 13:47 (Hand No.2); May 2, at 16:50 (Hand No. 4). 47

This has got to be Pokerstars at its best. Not only do these hands strikingly defy statistics in terms of the chance of dealing such hole cards, but, when looking closer at the hands and their unfolding (images on the right), all that has been so vehemently and frequently criticized online in regard to Pokerstars suddenly and vividly comes to light: minuscule probabilities materialized, the most improbable flops, the "coolers" meant to keep players in the game and bet massively, the sensational reversals of situations of turn and river, again defying statistics, etc. Take for instance the first hand captured: had I stayed in the 48

game, despite my pre-flop equity of only 20%, I would have won the hand by making the set on the turn (a roughly 8.4% probability of hitting it on either of the last two streets, respectively 4.65% of doing it specifically on the turn!). The second hand is similar: having started from a 15% initial equity, I would have won again the hand had I stayed in the game, by hitting a straight on the river (a 4-outer, to be more exact). Finally, the "winner" is probably the fourth hand: not only are 3 of the 6 players being dealt pocket pairs, with me smart enough to fold pre-flop, but the flop bears a set vs. set situation of Queens vs. Jacks! And if you think that might have been an awkward, but statistically still possible, exception, think again! 59 showdowns that have been reached when I held pocket 3s were also enough to discover an equal number of 4 triple pair cases (meaning 6.78%), and this in relation only to those hands that did reach showdown. The aggregated image covering all triple pair situations that I discovered at showdowns is captured in the table below. Table. Triple pocket pair situations recorded at showdowns:

Now, obviously, the overall percentage of 1.74% is inferior enough to the normal, expected one, of 5.75%. This, however, does not actually mean that by some miraculous happening, the occurrence of triple pair situations on Pokerstars would be lower than normal. Quite the contrary, in fact, when looking closer at the table. Firstly, it can easily be noticed that no less than 9 of the total 17 triple pairdisplaying showdowns, meaning 52.9% of them, involved a player with pocket Aces. This in turn does also not mean that Aces' frequency of dealing would be higher than normal, but only, and as simply as possible, that it is only when holding an extremely strong pair, actually the best hand in the Hold'em game, that players will not fold and instead go all the way to showdowns. It is exactly because the players engaged had Aces in their pocket that these cards' frequency is so high among the selected showdowns. With lower pocket pairs dealt, 49

people will naturally venture into showdowns way more seldom, folding along the way, especially if they did not hit a set or better on the flop, to a much higher degree. After all, this analysis covers real cash games, played on real money, which makes it only reasonable to attribute rationality to the players involved. This means inclusively that, when starting with a weak pocket pair, and not hitting an advantageous board, and additionally confronted with the opponents betting, they will be folding. Starting with this logic, and assuming, as if we were Pokerstars' attorney, that in no initial triple pocket pair situations did the players with AA fold before showdown 46, in relation to which 9 triple pair situations have been recorded, then, by extrapolation, we can estimate grosso modo that, corresponding to all 13 possible pocket pair hands existent, there should actually have been (13*9)*2 = 117 such instances of triple pairs among the showdowns, meaning 117*100/975 = exactly 12% of all hands taken to showdowns. This new figure is not only significantly higher than the mathematically normal frequency of 5.75% (meaning more than double), but also, judging in contextualization with all the dozens of items analyzed throughout this investigation47, also extremely plausible for the Pokerstars platform. Secondly, a closer look at the table reveals two cases in which two of the three pocket pairs were of equal rank: 99 vs. 33 vs. 33 and 55 vs. TT vs. 55. Then again, the probability of one single such situation is, as already indicated, a mere 0.185% in relation to a player's series of being dealt pocket pair hands. Well, if this recorded case represents that percentage, than it means, by extrapolation, that the total number of showdowns needed for these two events to be normal should have been 2*100/0.185 = 1081.08, which, again, is absurd, since the total number of showdowns that I have reached when holding a pocket pair was only 975. And, again, no, one cannot take into consideration the total number of pocket pair hands that I have been dealt since it can easily be counter-argued that among the hands folded pre-showdown there have also been cases of triple pair situations. The conclusion is as recurrent as redundant: the "cooler"-type situations on the Pokerstars platform occur significantly more often than in a genuinely random environment, and, no matter which technical method or argument or perspective is chosen, the recorded anomalies cannot be attributed to any randomness. They only represent the visible result of a deliberate, business-motivated, strategy of the poker platform.

II.4 An experiment: (open-) limp-check-calling with pocket pairs all the way to showdown Essentially when dealing with a situation such as the analyzed here, i.e. an online poker platform suspected to use a non-random software, there is, actually, one single definitive and irrefutable, but unfortunately impossible to apply method to prove that something is wrong48. This method would suppose that all the players should remain at the table every hand all the way to showdown, for a sufficient number of hands, in order to thus exhaustively and reliably test the randomness of the dealing algorithm. Surprisingly or not, this single usable option is one unavailable on Pokerstars, not even on play money: you cannot simply take 5 friends, open a table and just sit together till showdown, so you can test the statistics. There isn't, simply, such an option available, as one cannot open such a table. 46

Although, obviously, this also happens in reality depending on the board cards and the opposition's betting manner), and knowing AAs represent only one of the total 13 possible pair hands (or combinations). 47 From the frequency of heads up AA vs. KK situations, which can also offer an extrapolation starting point, all the way to the frequency of set vs. set situations on the flop. 48 Other than, obviously, having access to the cards-dealing software.

Given this insurmountable obstacle, I came up with the possibly second best method: for an entire 2,573 hands played in the timeframe May 26, 17:40 - May 27, 14:48 EET, at cash games, zoom format, 6-players rings, 1/2c stakes, each time I was dealt a pocket pair, I (open-) limped and simply check-called on all streets, with the sole purpose of dragging my opponent to showdown, in order to test two things: the recorded frequency of pair vs. pair encounters; the actual, measured, general win rate of pocket pairs in comparison to their mathematical, expected, win rate in the given game circumstances. With the second issue discussed later on in this study, the following addresses the first one. Table. Hands' outcomes when I held pocket pairs during the experiment:

Opponent's hole cards 1 2 3 4 5 6 7 8 9 10 11 12 13 My PP QJo 54o T7s KJo KQs KTo 22/T6o AA QQ 88 66 106o ATs/KQs K5o A6o J6o A4s ATo QJo KK AA/JJ AJo KJo K4s A9o 84o 97o A5s KJs 43o QQ 44 55 82s A7s 95o 72o JJ AA AQo Q7s Q9s ATo 42s AKo J9s TT 33 66 AJs JTo A8o AKs A3o 65s A3s Q4o 98s/87s A3s J4o T7o 99 88 J9o AJo/76s QTs AQs AKo AJs J6o KTs/K3o A3s 62o 99/K10s 88 44 44 108o AKo A8s 83o AKo T8s KQo 77 66 AQs A8o ? (BB)* 86o K7o A8s Q2o 75s Q9o KQo Q9o T8s 66 88 42o A6o 95s T6o T3o 54s/84s A6s 76o A4s A9o J2o 55 44 KJo 87o AQo Q7o KQo 108s 74s K8s 87s/T8s 97o 44 AA 87s AJs A9o 97o 33! AA 99 TT QQ 86o ATo 43s J7s 22! JJ 99 77 * All my opponents folded pre-flop, with me in the big blind position. Shadings: green = I won; pink = I lost; blue = split pot

AQo

AKs

Despite my (non-)betting trick meant to not dissuade any participation in the game, it is more than obvious I still haven't managed to persuade each player who held a pocket pair to walk along me till showdown. Players with, say, 22s or 33s might have folded pre-flop in front of me, not opening with such cards from early positions; others probably folded their pocket payers behind me, when confronted with an intermediary opponent's 3-bet, etc. Yet, this method provides a slightly better approximation of the pair vs. pair situations' frequency on Pokerstars. True, on the overall, nothing seems suspicious: Out of the 131 times I held a pocket pair, at showdown I encountered an opposition non-equal pair only 24 times, meaning in 18.3% of the cases, which is still significantly below the normal, expected, frequency of 26.13%. Thus, the findings are at first glance inconclusive.

But, as said, my method still could not ensure each pocket pair holder coming along to showdown. And in this regard, going back to the 2s and 3s pocket pairs mentioned above, they are actually those pairs providing the best approximation of the frequency sought: when I held 2s or 3s, my opponents, if they had any pairs, would have had better ones, pairs which they wouldn't have folded that easily, thus increasing the chance of reaching showdown. And the corresponding numbers speak for themselves: in no less than 46.7% of the cases (7/15, that is), I was dealt pocket 2s or 3s, at least one of my opponents, the one who stayed till showdown, also had a (different rank) pocket pair. This percentage is significantly higher than the normal, expected, frequency of 26.13 associated with non-equal pair vs. pair encounters in 6-players games. Extremely interesting in regard to this 46.7% figure, if we were to try another approximation in a completely different manner, and extrapolate on the basis of the cases in which, in all the 51,989 cash games played by me, I held Aces and encountered opposition Kings, knowing that the probability of such an event is 2.425%, and knowing that the normal pair vs. pair encounters' frequency is 26.45%, we would reach a necessary absolute frequency of Aces vs. any other pocket pair of 26.45*100/2.425 = 109.07 cases. Relating this to the total 232 hands of Aces that I have been dealt, it would mean that in 109.07*100/232 = 47.01% of all the cases I held Aces, there must have been at least another player at the table who had also been dealt a pocket pair. The difference between the two figures reached via separate methods, 46.7% in the first case vs. 47.01% in the second, is spectacularly small, both methods converging towards the conclusion that, on the Pokerstars platform, the frequency of pair vs. pocket pair situations is at a minimum 1.75 times higher, and deliberately so, than the normal, expected one in a genuinely random, probabilistic environment.49 However, still not at peace with being incapable of getting my opponents to showdown, it crossed my mind that in Pokerstars play money games people usually play significantly more loose, since no real money is involved, thus increasing the chances of reaching showdown. Thus, on the same evening of May the 27th, I ended the experiment and jumped onto play money tables. Since there is absolutely no reason to suspect that the platform uses one type of RNG in real money games and another one in play money games 50, it seemed like a feasible plan. So I picked up the lowest stakes level and played the same strategy of open-limping and just check-calling whenever I was being dealt a pocket pair. It worked; play money players are indeed looser, calling and opening with a significantly larger range of hands, so that I had the opportunity of taking all my pocket pairs to showdown. Well, after not even 150 hands played, I had already been dealt pocket pairs 11 times. In no less than 6 of these cases (that is a staggering 54.5%, but, remarkably, a figure not so distant from the overall 46.7-47.0% reached above!), at showdown I discovered one of my opponents also having a pocket pair: one time I was dealt Aces, and bumped directly into Kings, to my adversary's despair; one time I had Kings, and discovered an opponent with pocket 6s; my single time I was dealt Jacks was enough to bump into pocket 9s; of the two times I had tens, 49

No matter how you approach the findings within the experiment, they converge towards the same conclusion line, i.e. the significant over-representation of pocket pair duels. Let's return for a last time to the experiment, only this time from another angle, specifically focusing on those cases in which I encountered, distinctively, a pocket over-pair. Thus, I bumped for instance into Aces (never mind the simultaneous Jacks) when holding pocket Kings, despite the probability of such a scenario, in a 6-players ring, being a mere 2.425%. This means out of 50 KK hands played, I will / should encounter Aces 1.21 times - well, in my experiment, this happened, though I have played Kings only 9 times. If also taking into account the Jacks my second opponent had at the same hand, the probabilities go beyond ridiculous. And the same applies to Jacks: even though the probability of encountering an opposition over-pair is a mere 7.2% (meaning roughly one time for every 13.9 pair of Jacks played), it happened for me despite having played Jacks only 5 times! 50 If some Pokerstars representative would try claiming that there are two different RNGs used, depending on the real- or play- money nature of the games, it would, if you think about it, be the equivalent of shooting themselves in the foot.

one time I encountered Aces; three times I had 7s, and in one of them I hit an opponent with Queens; finally, having played pocket 4s one time was enough to bump, again, into Aces. You get the general idea. I repeat: in my estimation, based on combined 110,000+ hands played on both real and play money on Pokerstars, and on other thousands I have not participated in, but only observed, and using multiple separate methods of measurement and eventual extrapolation, I feel it is safe enough to estimate that the real frequency of pair vs. pair encounters in 6-players game is at least (i.e. a measured) 1.75, but possibly (i.e. remeasured, plus estimated by various methods) even 2.1 times, higher51 than the normal 26.45% aggregated one and than the 2.425% specifically-targeted one. As a final remark, I think it is crucial to specify that, based on multiple measurements engaged, my conviction is that these inflated frequencies recorded on the poker platform are not the sole consequence of players being dealt pocket pairs more frequently, but the result of an additional software scheme that separately and increases the likelihood of such encounters.

II.5 Summary of the findings Covering all the 55,320 hands that were dealt to me in both cash and tournament games, the general statistical analysis of the hole cards' distribution on the Pokerstars platform, as the first step of this chapter's analysis, failed, aside from some minor exceptions, to discover any significant anomalies in relation to an expected, mathematically normal, Gaussian, distribution, which would prima facie suggest that the platform's RNG deals the cards in a truly random manner. Even when starting with the second step of the analysis, namely a targeted analysis of the suited hole cards' distribution nothing suspicious appeared within my scope of observation, quite the contrary. Thus, all the general statistical indicators employed found recorded values that are almost perfectly overlapping the expected, normal, ones, with some of the differences even placed at the third decimal's position! Thus, overall, meaning at a general and superficial look, the distribution recorded on the poker platform is as normal as possible, and actually, and arguably, the most normal one could imagine / the most normal one has ever seen in nature! However, my52 educated guess that the software somehow manages to always provide normal average values for the general parameters at the end of the day (despite brutally intervening not parallel to, but flagrantly against randomness) has been satisfactorily confirmed when narrowing and deepening the analysis in four exploratory directions. Firstly, at an intermediary level of segmenting the 169 hands to be possibly dealt in the game, one easily notices that, in their decreasing order of their value in the game: pocket pairs are dealt the most frequently in relation to both their expected frequency and other possible hole cards; the next ones, also above the normal frequency, are the suited hole cards, which, on the average, make up the best hands other than pairs; off-suite hole cards, the weakest among the three groups, are being dealt below their expected frequency. Whereas, judged in itself, meaning isolated, this overlapping of power ranking and frequency of distribution might be still be a coincidence, things get a bit more difficult to justify, if walking in Pokerstars' shoes, when discovering narrowed down statistical anomalies, such as an indisputable correlation (-0.76), along the suited connectors series, between their power ranking in 6-players games and the frequency of the RNG dealing them 51

This is just a "guesstimate" based on the same 110k+ hands played, a speculation that I did not engage to test, but it's possible that the higher one's pocket pair is, the higher the chance of bumping into another high pocket pair, while the corresponding frequency for lower pairs might be smaller. 52 And not only mine, as one can easily notice when using online search engines to read out about Pokerstars' alleged irregularities.

to me. Additionally, there is also some discernible relation between the best hierarchical groups of hole cards (arranged according to their win plus split rate) and their dealing frequency, although, overall, meaning along all the 169 hands in the game, the general correlation is, again, almost inexistent. The second confirmatory direction targeted separately the pocket pairs' distribution. In this case, all the statistical methods and indicators employed converge towards a definitive and categorical conclusion: the systematic over-representation of pocket pairs as hole cards cannot, by any means nor by any logical contortionism or "variance" invoking, be considered just an(other) "accident" and attributed to randomness; it substantially hints at a deliberate, premeditated, strategy of the poker platform, justified by the very way in which it makes a living of: rake. To this end, by dealing good, strong, hands, more frequently, it increases the "fun" for the players involved, which in turn may attract new players, which, subsequently and decisively, will only increase the rake to be collected by the platform. The third explored and confirmatory direction of analysis, directly related to the previous one, focused on the so-called "coolers" on Pokerstars. And there have been thousands of people worldwide that have been complaining on the Internet about these "coolers", starting from the classic AA vs. KK duels, all of them significantly more frequent the normal (read: almost twice as frequent), and all the way up to quads vs. quads duels. And indeed some of the "accidents" happening on Pokerstars are truly grotesque judging by their minuscule probabilities, all clearly indicating anything but randomness. Among the extremely selective chosen examples from this chapter, one can find situations such as  me holding Aces and running into opposition Kings 1.8 more frequent than normal (and that is a minimum, assuming no KK holder ever folded a hand against my Aces before showdown!), two of the cases being consecutive AA hands played by me, which, in order to be normal, would require me to have played 159 hands, whereas the real, actual, number has only been 131;  three people at a 6-players table all having hearts-suited hole cards, with the five board cards giving them the hearts flush - a probability of 0.0298% of all the suited hands played;  an AA vs. KK vs. KK situation at a 6-players table - again, a infinitesimal probability;  a KK (me) vs. AA vs. JJ situation at the 6-players table, with me flopping a quads of Kings, an event that, in order to be considered statistically normal, would have required me to have played some 85 million hands, whereas my total combined cash + tourney games has barely reached 55.3k!;  a equally sensational 4s vs. As quads at a 4-players table, something which, from the perspective of the 4s-holding player, should happen to him once in every 81 million hands dealt! One of Pokerstars' favorite excuses over time has been that anomalies such as the ones exemplified happen in online poker for the sole reason that one is playing significantly more hands than in any live form of poker. Well, I would say: go ahead and convince me that, in regard to the third example, I did indeed play 85 million hands! Or try explaining to the poor guy holding the pocket 4s in the last example why that "accident" happened to him, even though one could safely bet he did not play that necessary 81 million hands! And, rhetoric aside, the crucial problem, as explained, is not that such things do happen, since they are (indeed, extremely) improbable, but nevertheless possible. After all, people get hit by meteorites, others win the big lottery, while others suffer from a 1 in 10 million rare disease, etc. The problem, and this is where Pokerstars' representatives have a tremendously difficult, or rather impossible, task in justifying themselves, is that all of these beyond-exceptional hands have one thing in common: all of them lead naturally, surely, implacably, to the same outcome - an increase of the pots by betting, raising and reraising, which in turns provides the platform with a higher rake to collect! 54

The fourth direction, a spin-off of the third, has extensively addressed the pocket pair vs. pair situations as another aspect all to frequent on the Pokerstars platform, and, as a subtype of the more broadly delineated "coolers" also ensuring a rake increase. Measured and re-estimated by various separate methods, and re-confirmed by a definitely unusual limpcheck-call experiment that I engaged in over 2.5k hands played, the frequency of pair vs. pair duels on the investigated poker platform is a measured minimum of 1.75 times, respectively an estimated possible maximum of 2.1 times higher than the normal frequency to be expected in a genuinely randomness-characterized environment. Finally, the same conclusion has only been reconfirmed when specifically analyzing the frequency two distinct phenomena, namely equal pair encounters and triple pair situations in 6-players games, with both types of events occurring, based on a set of extrapolations, twice more often than normal. Correlation, regression, extrapolation, standards deviation and normality of distribution, empirical observation, individual and cumulated absolute and relative frequencies, reductio ad absurdum, timeframe segmenting, - you name it, I've tried it. All of these methods, indicators, parameters, and techniques employed in this chapter across the four mentioned exploratory analysis directions converge towards one single categorical conclusion: the way the hole cards are dealt by Pokerstars supposedly "random" numbers generator is anything, but random. Moreover, as the next chapters of this investigation shall also demonstrate and gradually reconfirm, what happens on the platform is not the visible effect of some unintended error, of some "glitch" within programming, but the expression of a deliberate, premeditated, and systematically applied policy of the platform's owners.

III. THE FLOPS: when you take the bait The present chapter investigates the alleged randomness of the flops I have encountered on the Pokerstars platform, by testing two fundamental hypotheses: a) the so-called "second shuffle" hypothesis, which states that the flops are not really random, but rather directly related to the exact hole cards of the players that have remained at the table beyond pre-flop, meaning that, after the initial hole cards dealing, there could be an additional, second, and severely non-random "shuffle", and b) the "leveling the field", a.k.a. "balancing the odds" hypothesis, which also suggests nonrandomness, but specifically looking at how the favorite vs. underdog duels on the platform illustrate a higher than mathematically expected win rate for the underdog. To this end, after a general analysis as a first step, this chapter continues with a second step in the form of a case study based on all the hands (within the cash games) that I have been dealt pocket Queens. This case study shall re-target the two hypotheses, in a preliminary and partial confirmation, in order for the third part of this chapter to deepen the analysis of Pokerstars' flops across four analytic items: i.) flopping a set or better when starting with pocket pairs; ii.) flopping a flush draw or better when starting with suited hole cards; iii.) flopping at least one pair (a cumulative category, segmented at the right time) when starting with non-paired hole cards; iv.) flopping straight draws or straights when starting with middling connectors as hole cards, regardless whether they are suited or not.

III.1 An overall look at the flops on Pokerstars Much has been debated online over the years about the implausible frequency of situations in which either on the turn or, especially so, on the river, equities are reversed in spectacular turns of the tide, which has even earned Pokerstars one (other) nickname "Riverstars". It needs to be said that none of the dozens of items (each containing from tens up to thousands of cases) I have analyzed confirm this speculation; wherever I looked, from all-ins engaged on turn or river and up to hands that reached showdown step-by-step, probabilities on the two streets are generally and at least satisfactorily respected provided the sample of hands is representative enough. Indeed, in cash games and tournaments as well, and regardless of the game format or stakes size, the Pokerstars platform does host some "accidents", some of them truly "horrendous", I'd say, such as, for instance, among the hands I personally played: ď&#x192;&#x2DC; the "classic" of hitting a 2-outer on the river (2 useful cards out of a deck of 44 cards, meaning a probability of 4.55%) exactly when you need it the most:

ď&#x192;&#x2DC; or reversing a 2% equity on the flop, against a flopped set, by hitting a backdoor straight on turn and river:

ď&#x192;&#x2DC; or winning the hand despite a... 1% equity on the flop, against a player who had flopped directly a full house(!):

ď&#x192;&#x2DC; or, finally, achieving the same performance in terms of equity reversal, but in a different manner:

In spite of such dramatic reversals of situations, easily etched in the memory of players, especially if they happened to their detriment, as said, over volumes of hands large enough, probabilities seem to be respected on Pokerstars at least to a satisfactory degree. For instance, within the total 51,989 hands that I played in cash games, I isolated those hands when, simultaneously, a.) I have been dealt pocket pairs; b.) the hands reached showdown; c.) 57

on the turn I was either a 95.45% favorite to win the hand, or the 4.55% underdog (meaning, simplified, I needed any of only two outs to win the hand). The results are these: ď&#x20AC; of the 112 hands in which, on the turn, I was a 95.45% favorite to win, I won 104, meaning 92.86%. The percentage is a little smaller than the expected one, but on the one hand not significantly so (minus 2.7% variation of the normal figure), and on the other hand it could indeed be argued that, whereas I personally played thousands of hands in which I deliberately stayed till showdown, despite strongly suspecting I was the underdog or even that I would not win, my opponents should have played more win-oriented. This would mean there could have been cases in which my 95% advantage on turn did hold, meaning I had mathematically won the hand(s), but my opponents folded on the river, just before showdown. If this was indeed the case, not the same argument could in turn be made in relation to ď&#x20AC; the 194 hands when I was the 4.55% underdog on the turn, out of which I won 9, meaning 4.64%, which is even slightly above the expected frequency. Thus, as a general conclusion, probabilities at least on the turn, before the river, seem to be generally respected on the Pokerstars platform. The actual issue, or better said, this other issue, with Pokerstars has nothing (or, better said, not so much) to do with the turn or the river cards, but, and this might come as a surprise, with the flops! The reason why, to the best of my knowledge, no one has focused on them so far, has probably to do with two reasons. Firstly, sensational twists on the river such as hitting a two-outer (as mentioned, a probability of less than 5%), especially when to one's detriment and direct financial loss, will naturally be more easily remembered. Secondly, unlike the river, the flop contains not one card, but three, which makes the mathematics associated with it significantly more complex and correspondingly more difficult to grasp, let alone master. Whereas in a heads-up situations, once the players have reached the turn, there are only 44 mathematically possible river cards53 and only one card to be dealt, a flop, even if considering the opponent's hole cards as known, may contain no less than 17.296 cards combinations54. Hence, "guesstimating" or actually calculating the probability of a specific river card is one thing, whereas doing the same thing for one particular flop is a completely, and significantly more difficult, matter. As an additional factor, only a fraction of the hands are played, by at least two players, till showdown, thus allowing us to see in what manner the flop cards related to the hole cards of the players at the table. Hence, even if something about the flop would be peculiar, or significantly improbable, for the average player who is not a mathematician and does not possess a software to store the played hands (and indicate equities) it would be extremely difficult to notice it. And indeed there is something... "fishy" about the flops on Pokerstars, many of which cannot, by any overstretch of the argument, be attributed in a compensating-type explanation not even to those flops that weren't continued till showdown, flops which, in the speculative explanation, are claimed to have compensated any anomaly among the hands that did reach showdown. The underlying logic would be something like: "True, your flops have illustrated some exceptional probabilities, but all the other flops, among the hands that did not reach showdown, were 'normal', so that, on the overall, statistics are respected, and things are not uncommon!". Again, I categorically reject such a possible pseudo-argument. Allow me to first remind the reader of two examples already given in the previous chapter, and then offer another set of five examples, plus a bonus, from among the hands that I played. The two previously listed examples are the 0.039% probable J3K flop at a table with three opponents holding in their pockets KJo, JJ, respectively 33. The second, that one lucky hand of mine, was the triple pair situation where I held pocket Kings, suddenly woke up 53 54

Assuming, of course, both sets of hole cards are known. Respectively 19,600 if the hole cards of the opponent are unknown.

with one opponent having Aces, the other Jacks, only for me to hit the jackpot of flopping Kings quads. Leaving aside the already tiny pre-flop probability of three players having exactly those pocket pairs, the probability of such a flop, in that given situation, was a mere 0.29%, which means that in a 6-players game, such an event will happen once in every 345 hands of KK that I am dealt (whereas, of course, redundantly specified) I did not actually get 345 KK hands. Now let's get to another 5 + 1 new examples of flops that defy statistics and refute speculations, all of them selected from the subsample of hands in which I had been dealt pocket pairs, and all of them reinforcing the recurrent idea that Pokerstars will abundantly and deliberately provide players with statistically quasi-impossible flops, meant only to determine the players to stay in the game, bet high and thus increase the rake collected by the platform. The first flop, captured below, is already a shocker, and strikingly resembles the first of the two examples recounted above:

One opponent has a pocket pair, while his two opponents somehow manage to simultaneously flop two pairs each, starting from non-paired hole cards. Well, for those not really into math, let's simplify the explanation. Considering the 6 known hole cards of the players, of the total C(46, 3) = 15,180 flops possible, there are only 18 providing the above situation: 2 (any of the two Kings left in the deck) * 3 (any of the three 10s in the deck) * 3 (any of the three Jacks in the deck). Thus, the probability of such a flop, considering the starting pre-flop hole cards, is 18*100/15180, meaning a minuscule 0.118%! To formulate it differently: out of 1,000 such starting situations, a flop such as the one above will be dealt a little over one time! Obviously, I did not start form such a pre-flop position 1,000 times; there was only a total 3,146 times that I have been dealt pocket pairs, of which only 2,132 times did I get to play or at least see the flop. Still, it seemingly was enough for me to encounter such a statistical improbability! The second example is one equally spectacular, or actually even more so, considering it's probability to be dealt is slightly smaller, namely 0.104%!

Of the total - in this case - no less than 17,296 flops possible (C 48,2), only 2 (any of the two 6s in the deck) * 3 (any of the three 7s in the deck) * 3 (any of the three 8s in the deck), meaning only 18, could have generated a situation in which one player has a set, whereas the other has two pairs. And still, this exceptional case also happened! I guess that, bearing a 0.237% probability, the third flop used here as an example should be considered at least in comparison to the previous two, as being extremely likely?!

Out of 15,180 flops possible, I was apparently lucky enough to get exactly one of the only 18 that could simultaneously guarantee my two opponents the same Jacks trip! The fourth example is a hand when I was inspired enough to fold pre-flop, but my two opponents continued the hand all the way to showdown, so that the relation of the flop to their cards could be visualized:

The flop manages to simultaneously grant the players: a) a top-pair or at least what the AJs player might think is a top-pair; b) a set for the JJ-holding player, and c) another set, the lower one, for me, had I stayed in the game. Without going into further mathematical details, let's just state the probability of such a flop: 0.57%. If these superficially selected four cases weren't enough, the fifth example should, normally, close the case. The two screenshots below capture two clusters of hands of pocket 4s that I played in cash games on Pokerstars, as stored by the HM2 software, with my pairs in the second column from the left, while the fourth one indicates the board cards for that respective hands:

So, in a matter of less than 24 hours, between April 14, 17:23, and April 15, 14:57, there were three consecutive instances among the pocket 4s hands that I had been dealt that I flopped a set! Well, the per se probability of flopping a set is 11.17% (or 11.76% for the cumulative category "set or better", which includes sets, full houses, and quads). The conditioned probability of flopping a set or better three times in a row is a mere (0.1176)^3 = 0.16%! But that's not it; less than a month later, I managed (well, correction: Pokerstars' RNG actually managed) to do the same thing four times in a row55 (out of which one was a quads case), a chain of events with a conditioned probability of... 0.0019%!

The intermediary hand of 44s that did not reach the flop can obviously not be taken in account, since anything could have happened in regard to it.

That would mean that, in a genuinely statistically normal, meaning authentically random, environment, such a series of flops would be expected to happen (almost) once in every 50,000 hands of pocket pairs played!56 To put it differently: if I had played 50k pocket pairs, then would such an event be possibly regarded as statistically normal, if there were any doubts left, allow me to specify: the total number of all pocket pairs I was dealt over all the 51,989 hands in cash games on Pokerstars has been 3,146, out of which a mere 2,132 made it (by me personally playing or my opponents continuing them) to the flop! And still, such a astonishing exceptionality once again made its way through Pokerstars' RNG mechanics! As a parenthesis: similar things happen even when having non-paired cards; one can, for instance, flop two pairs 4 out of 5 consecutive times, which equals a probability of 2.02% * 2.02% * 97.98% * 2.02% * 2.02% = 0.0000163%, meaning an event expected to happen once in every 61.35 million non-paired hands played, where in reality the total number of flops I have played or seen when holding non-paired hole cards has been less than 20k! It is that simple: Pokerstars apparently has no hesitation whatsoever in providing you with the most extraordinarily improbable flops, all apparently in your favor, as long as you bet, raise, and re-raise, thus guaranteeing an increase of the rake it collects!

Or, by introducing the 5.88% probability of being dealt a pocket pair, it should happen once in every 1.12 million hands played (regardless whether I was dealt paired or nonpaired hole cards)! All this not even taking into account either a.) the probability of the first two flops in discussion (the inferior screen capture) containing the same value cards - K, 9, and 4, or b.) the probability that both of the first two hands ended up with me making a full house!

Finally, the promised bonus: my flopped royal flush, the best hand in poker!

For those not very familiar with probabilities, allow me to specify the one of an event such as the above: there are 1326 two-cards combinations possible in relation to a 52-cards deck that you can be dealt at the table. Out of those 1326, there are only 40 that will put you pre-flop in the necessary position to flop a royal flush draw, the made royal flush included: 4 (combinations) for each of these ten hands: AKs, AQs, AJs, ATs, KQs, KJs, KTs, QJs, QTs, JTs. That means that in only 100*40/1326 = 0.302% of all hands to be possibly dealt, one gets, in a first step, in the pre-flop position necessary to aim at flopping the royal flush. However, even if you are dealt one of those 40 combinations, there is the second step, even more improbable than the first one: only one of 19,600 possible flops (C50,3) will contain the targeted combination, meaning those specific 3 cards that will give you the royal flush; for instance, if you were dealt AJ of diamonds, the three flop cards must be the K, the Q and the T of diamonds, and none else. In conditioned terms, that means a probability of (40/1326)*(1/19600)*100 = 0.0001539% of flopping the royal flush or, in an alternative formulation, such an event will happen once in every 649,772 hands played! Well, apparently not on the Pokerstars platform, where my 51,989 hands played in cash games have proven to be more than enough for such a sensationally improbable event to occur! And no, that was not an "accident", although, under normal circumstances, such an event might, indeed, happen, in the sense that it is possible, though extremely improbable. If further proof for such a statement was needed, then I should add that, among the exactly 999 flops that I have reached in total when holding any of the 40 above-listed necessary combinations, I have additionally flopped a royal flush draw no less than 6 (six) times57, 57

Out of which one even turned into a royal flush (see above) - an in itself probability of 4.25%.

which, excluding the made royal flush, equals a recorded frequency of 6/998 = 0.601%, which is 2.5 times higher than the normal, expected one, of 0.239%!58

It seems that everywhere you look, the flops are exceptional in terms of probabilities. In order to encounter such unbelievable "miracles", or, depending on the perspective and outcome, terrible "misfortunes", I (and to an equal degree, your average player on Pokerstars) should have played hundreds of thousands of games, which, of course, is not the case. And, as repeatedly argued and stressed throughout this paper, the problem is not that such events do occur, but that all of them tend to happen in the same direction, i.e. directly corresponding to the hole cards of players remaining at the table and to their apparent benefit! More generally, such observed anomalies are simply the visible effect of an articulate and deliberate business strategy implemented by the poker platform, meant, among other things, to increase the fun, to deliver adrenalin and spectacle via all kinds of sensational hands and turnings of situations, in order to increase its player base and subsequently the rake it collects. To this end, on Pokerstars, as this entire study demonstrates, better cards will be dealt to you more often than normal, flops will contain enticing combinations more frequently, combinations such as sets, full houses or quads when starting with pocket pairs, or flush draws when starting holding two suited cards, or straight draws, trips and two pairs, when holding non-paired cards. Equally true, there will also be an abundance, again significantly above normal parameters, of the so-called "coolers" (e.g., pair vs. pair duels, straight draws vs. flush draws, flushes vs. sets, even quads vs. quads, etc.), in order to push players to bet and raise and thus increase the pot and the rake to be collected by the platform.

Or a recorded 0.700% vs. an expected 0.245% if the flopped royal flush is taken into account.

III.2 A case study: my pocket Queens In another distinct testing of this analysis' main hypotheses, the following isolates as a study case the sample of 249 pocket Queens that I have been dealt over the investigated timeframe in 51,989 cash-games. Of the total 249, - 95 I won without the hands being played (opponents folded pre-flop), - 67 have been played at least on flop, but were folded before showdown by either me (14) or my opponents (53), and - 87 have reached showdown.

III.2.1 The data commented The table below lists all the 87 hands that reached showdown, inventorying my hand, the opponents' hands, the flop cards along, when needed, a short comment, and the final outcome of the hand in terms of both me winning or losing, and the favorite (regardless whether it was me or the opponent) winning or losing. All probabilities listed in the table have been recalculated accordingly to the two set of hole cards known. In terms of the shadings in the extreme right column, green indicates that the pre-flop favorite won, red that he lost, while blue signifies a split pot, while "W", "L", and "S" indicate whether I won, lost or tied. No.

Date

My cards

Opp. cards

Flop

28.5 27.5

Time (EET) 16:23 13:08

1. 2.

QQ QQ

QT 34

J6A 8Q8

27.5

12:44

4. 5.

27.5 27.5

11:59 00:27

QQ QQ

27.5

00:10

26.5

23:33

26.5

22:37

Situation on flop (probability of hitting that situation of flop (if relevant)):

Opponent: flush draw + gut-shot straight draw (2.20%) Me: full house (0.76%) Me: set (11.17%) KJ 10 Q 3 Opponent: open-ended straight draw (2.13%) Opponent: one pair (33.63%) + backdoor flush / straight draw A5 725 Opponent: one pair (33.63%) + backdoor straight draw 79 389 Comment: with a 25% equity on flop, my opponent hits his second pair (7s) on turn and wins 55

4K8

W W L W

Me: set (11.17%) 48 76Q Opponent: gut-shot straight draw (12.49%) Comment: with a 15% equity on flop, my opponent hits the 5 on turn, making a straight, and wins 44

Outcome (for me): W W

5AK

L W

26.5

21:51

387

10. 11.

26.5 26.5

19:02 18:36

QQ QQ

4K KJ

K37 KJ7

12.

26.5

18:17

752

13.

26.5

16:14

922

14.

26.5

15:10

15. 16. 17.

25.5 25.5 23.5

18:25 13:31 13:14

QQ QQ QQ

18.

23.5

13:02

19.

23.5

20.

22.5

Comment: after the turn 7, with an equity of 14%, my opponents hits K on the river and wins Me: set (11.17%) AA JQT Opponent: gut-shot straight draw (28.10%) Comment: Despite a mere 18% pre-flop equity, I flop a set at the right time and win, with opponent missing both the straight and a set on turn or river (22% equity on flop) Opponent: gut-shot straight draw + backdoor flush draw T9 865 Opponent: pocket over-pair + gut-shot straight draw (1.48%) AA 342 AA

K23

379