(Part 1)
5
Chapter
Probability
Random Experiments Probability Rules of Probability Independent Events
McGraw-Hill/Irwin
Copyright Š 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
Random Experiments Sample Space • A random experiment is an observational process whose results cannot be known in advance. • The set of all outcomes (S) is the sample space for the experiment. • A sample space with a countable number of outcomes is discrete.
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Random Experiments Sample Space • For example, when Citibank makes a consumer loan, the sample space is: S = {default, no default} • The sample space describing a Wal-Mart customer’s payment method is: S = {cash, debit card, credit card, check}
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Random Experiments Sample Space • For a single roll of a die, the sample space is: S = {1, 2, 3, 4, 5, 6} • When two dice are rolled, the sample space is the following pairs: S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} 5A-4
Random Experiments Sample Space Consider the sample space to describe a randomly chosen United Airlines employee by: 2 genders, 21 job classifications, 6 home bases (major hubs) and 4 education levels There are: 2 x 21 x 6 x 4 = 1008 possible outcomes It would be impractical to enumerate this sample space. 5A-5
Random Experiments Sample Space • If the outcome is a continuous measurement, the sample space can be described by a rule. • For example, the sample space for the length of a randomly chosen cell phone call would be
S = {all X such that X > 0} or written as S = {X | X > 0} • The sample space to describe a randomly chosen student’s GPA would be
S = {X | 0.00 < X < 4.00} 5A-6
Random Experiments Events • An event is any subset of outcomes in the sample space. • A simple event or elementary event, is a single outcome. • A discrete sample space S consists of all the simple events (Ei): S = {E1, E2, …, En} 5A-7
Random Experiments Events • Consider the random experiment of tossing a balanced coin. What is the sample space? S = {H, T} • What are the chances of observing a H or T? • These two elementary events are equally likely. • When you buy a lottery ticket, the sample space S = {win, lose} has only two events. Are these two events equally likely to occur? 5A-8
Random Experiments Events
(Figure 5.1)
â&#x20AC;˘ A compound event consists of two or more simple events. â&#x20AC;˘ For example, in a sample space of 6 simple events, we could define the compound events A = {Music, DVD, VH} B = {Newspapers, Magazines}
These are displayed in a Venn diagram:
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Random Experiments Events • Many different compound events could be defined. • Compound events can be described by a rule. • For example, the compound event A = “rolling a seven” on a roll of two dice consists of 6 simple events: S = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
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Probability Definitions â&#x20AC;˘ The probability of an event is a number that measures the relative likelihood that the event will occur. â&#x20AC;˘ The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 < P(A) < 1
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If P(A) = 0, then the event cannot occur.
If P(A) = 1, then the event is certain to occur.
Probability Definitions • In a discrete sample space, the probabilities of all simple events must sum to unity: P(S) = P(E1) + P(E2) + … + P(En) = 1 • For example, if the following number of purchases were made by
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credit card:
32%
P(credit card) = .32
debit card:
20% Probability
P(debit card) = .20
cash:
35%
P(cash) = .35
check:
18%
P(check) = .18
Sum = 100%
Sum = 1.0
Probability • Businesses want to be able to quantify the uncertainty of future events. • For example, what are the chances that next month’s revenue will exceed last year’s average? • How can we increase the chance of positive future events and decrease the chance of negative future events? • The study of probability helps us understand and quantify the uncertainty surrounding the future. 5A-13
Probability
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Probability What is Probability? Three approaches to probability: Approach Empirical
Example There is a 2 percent chance of twins in a randomlychosen birth.
Classical
There is a 50 % probability of heads on a coin flip.
Subjective There is a 75 % chance that England will adopt the Euro currency by 2010. 5A-15
Probability Empirical Approach â&#x20AC;˘ Use the empirical or relative frequency approach to assign probabilities by counting the frequency (fi) of observed outcomes defined on the experimental sample space. â&#x20AC;˘ For example, to estimate the default rate on student loans: P(a student defaults) = f /n = number of defaults number of loans 5A-16
Probability Empirical Approach â&#x20AC;˘ Necessary when there is no prior knowledge of events. â&#x20AC;˘ As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.
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Probability Law of Large Numbers • The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one. • Flip a coin 50 times. We would expect the proportion of heads to be near .50. • However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.). • A large n may be needed to get close to .50. • Consider the results of 10, 20, 50, and 500 coin flips.
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Probability (Figure (Figure 5.2) 5.2)
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Probability Practical Issues for Actuaries â&#x20AC;˘ Actuarial science is a high-paying career that involves estimating empirical probabilities. â&#x20AC;˘ For example, actuaries - calculate payout rates on life insurance, pension plans, and health care plans - create tables that guide IRA withdrawal rates for individuals from age 70 to 99
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Probability Practical Issues for Actuaries Challenges that actuaries face: - Is n â&#x20AC;&#x153;large enoughâ&#x20AC;? to say that f/n has become a good approximation to P(A)? - Was the experiment repeated identically? - Is the underlying process invariant over time? - Do non-statistical factors override data collection? - What if repeated trials are impossible? 5A-21
Probability Classical Approach • In this approach, we envision the entire sample space as a collection of equally likely outcomes. • Instead of performing the experiment, we can use deduction to determine P(A). • a priori refers to the process of assigning probabilities before the event is observed. • a priori probabilities are based on logic, not experience. 5A-22
Probability Classical Approach â&#x20AC;˘ For example, the two dice experiment has 36 equally likely simple events. The P(7) is number of outcomes with 7 dots 6 P( A) = = = 0.1667 number of outcomes in sample space 36 â&#x20AC;˘ The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice: 5A-23
Probability Subjective Approach • A subjective probability reflects someone’s personal belief about the likelihood of an event. • Used when there is no repeatable random experiment. • For example, - What is the probability that a new truck product program will show a return on investment of at least 10 percent? - What is the probability that the price of GM stock will rise within the next 30 days? 5A-24
Probability Subjective Approach â&#x20AC;˘ These probabilities rely on personal judgment or expert opinion. â&#x20AC;˘ Judgment is based on experience with similar events and knowledge of the underlying causal processes.
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Rules of Probability Complement of an Event â&#x20AC;˘ The complement of an event A is denoted by Aâ&#x20AC;˛ and consists of everything in the sample space S except event A.
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Rules of Probability Complement of an Event • Since A and A′ together comprise the entire sample space, P(A) + P(A′ ) = 1 • The probability of A′ is found by P(A′ ) = 1 – P(A) • For example, The Wall Street Journal reports that about 33% of all new small businesses fail within the first 2 years. The probability that a new small business will survive is: P(survival) = 1 – P(failure) = 1 – .33 = .67 or 67% 5A-27
Rules of Probability Odds of an Event • The odds in favor of event A occurring is P( A) P( A) Odds = = P( A ') 1 − P( A)
• The odds against event A occurring is P( A′) 1 − P( A) = Odds = P ( A) P( A)
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Rules of Probability Odds of an Event • Odds are used in sports and games of chance. • For a pair of fair dice, P(7) = 6/36 (or 1/6). What are the odds in favor of rolling a 7? P(rolling seven) 1/ 6 1/ 6 1 = = = Odds = 1 − P(rolling seven) 1 − 1/ 6 5 / 6 5
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Rules of Probability Odds of an Event • On the average, for every time a 7 is rolled, there will be 5 times that it is not rolled. • In other words, the odds are 1 to 5 in favor of rolling a 7. • The odds are 5 to 1 against rolling a 7. • In horse racing and other sports, odds are usually quoted against winning. 5A-30
Rules of Probability Odds of an Event â&#x20AC;˘ If the odds against event A are quoted as b to a, then the implied probability of event A is: a P(A) =
a+b
â&#x20AC;˘ For example, if a race horse has a 4 to 1 odds against winning, the P(win) is a 1 1 = = = 0.20 or 20% P(win) = a + b 4 +1 5 5A-31
Rules of Probability Union of Two Events (Figure 5.5)
• The union of two events consists of all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A ∪ B or “A or B”). ∪ may be read as “or” since one or the other or both events may occur. 5A-32
Rules of Probability Union of Two Events
• For example, randomly choose a card from a deck of 52 playing cards. • If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q ∪ R? • It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways).
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Rules of Probability Intersection of Two Events • The intersection of two events A and B (denoted A ∩ B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B. ∩ may be read as “and” since both events occur. This is a joint probability. 5A-34
(Figure 5.6)
Rules of Probability Intersection of Two Events
• For example, randomly choose a card from a deck of 52 playing cards. • If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q ∩ R? • It is the possibility of getting both a queen and a red card (2 ways).
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Rules of Probability General Law of Addition • The general law of addition states that the probability of the union of two events A and B is: P(A ∪ B) = P(A) + P(B) – P(A ∩ B) When you add the P(A) and P(B) together, you count the P(A and B) twice. 5A-36
A and B
A
B
So, you have to subtract P(A ∩ B) to avoid overstating the probability.
Rules of Probability General Law of Addition • For the card example: P(Q) = 4/52 (4 queens in a deck) P(R) = 26/52 (26 red cards in a deck) P(Q ∩ R) = 2/52 (2 red queens in a deck) P(Q ∪ R) = P(Q) + P(R) – P(Q ∩ Q) Q and R = 2/52
Q 4/52
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R 26/52
= 4/52 + 26/52 – 2/52 = 28/52 = .5385 or 53.85%
Rules of Probability Mutually Exclusive Events • Events A and B are mutually exclusive (or disjoint) if their intersection is the null set (φ) that contains no elements.
If A ∩ B = φ, then P(A ∩ B) = 0
Special Law of Addition • In the case of mutually exclusive events, the addition law reduces to:
P(A ∪ B) = P(A) + P(B)
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Rules of Probability Collectively Exhaustive Events â&#x20AC;˘ Events are collectively exhaustive if their union is the entire sample space S. â&#x20AC;˘ Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events.
Warranty
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No Warranty
For example, a car repair is either covered by the warranty (A) or not (B).
Rules of Probability Collectively Exhaustive Events â&#x20AC;˘ More than two mutually exclusive, collectively exhaustive events are polytomous events. events Cash
Debit Card Credit Card
Check
For example, a Wal-Mart customer can pay by credit card (A), debit card (B), cash (C) or check (D). 5A-40
Rules of Probability Categorical Data
â&#x20AC;˘ Categorical data can be made dichotomous (binary) by defining the second category as everything not in the first category.
Categorical Data
Binary (Dichotomous) Variable
Vehicle type (SUV, sedan, truck, motorcycle)
X = 1 if SUV, 0 otherwise
A randomly-chosen NBA playerâ&#x20AC;&#x2122;s height
X = 1 if height exceeds 7 feet, 0 otherwise
Tax return type (single, married filing jointly, married filing separately, head of household, qualifying widower)
X = 1 if single, 0 otherwise
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Rules of Probability Conditional Probability • The probability of event A given that event B has occurred. • Denoted P(A | B). The vertical line “ | ” is read as “given.” P( A ∩ B) P( A | B) = P( B)
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for P(B) > 0 and undefined otherwise
Rules of Probability Conditional Probability • Consider the logic of this formula by looking at the Venn diagram. The sample space is P( A ∩ B) P( A | B) = restricted to B, an event P( B) that has occurred. A ∩ B is the part of B that is also in A. The ratio of the relative size of A ∩ B to B is P(A | B). 5A-43
Rules of Probability Example: High School Dropouts • Of the population aged 16 – 21 and not in college: Unemployed
13.5%
High school dropouts
29.05%
Unemployed high school dropouts
5.32%
• What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout? 5A-44
Rules of Probability Example: High School Dropouts • First define U = the event that the person is unemployed D = the event that the person is a high school dropout P(U∩D) = .0532 P(U) = .1350 P(D) = .2905 P(U ∩ D) .0532 = = .1831 P(U | D) = P( D) .2905
• P(U | D) = .1831 > P(U) = .1350
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• Therefore, being a high school dropout is related to being unemployed.
Independent Events • Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A). • To check for independence, apply this test: If P(A | B) = P(A) then event A is independent of B. • Another way to check for independence: If P(A ∩ B) = P(A)P(B) then event A is independent of event B since P(A | B) = P(A ∩ B) = P(A)P(B) = P(A) P(B) P(B) 5A-46
Independent Events Example: Television Ads • Out of a target audience of 2,000,000, ad A reaches 500,000 viewers, B reaches 300,000 viewers and both ads reach 100,000 viewers. 300, 000 500, 000 P( B) = = .15 P ( A) = = .25 2, 000, 000 2, 000, 000 100, 000 P( A ∩ B) = = .05 2, 000, 000
What is P(A | B)?
P( A ∩ B) .05 = = .30 P( A | B) = .3333 or 33% P(B) .15
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Independent Events Example: Television Ads • So, P(ad A) = .25 P(ad B) = .15 P(A ∩ B) = .05 P(A | B) = .3333 • Are events A and B independent? • P(A | B) = .3333 ≠ P(A) = .25 • P(A)P(B)=(.25)(.15)=.0375 ≠ P(A ∩ B)=.05
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Independent Events Dependent Events • When P(A) ≠ P(A | B), then events A and B are dependent. • For dependent events, knowing that event B has occurred will affect the probability that event A will occur. • Statistical dependence does not prove causality. • For example, knowing a person’s age would affect the probability that the individual uses text messaging but causation would have to be proven in other ways. 5A-49
Independent Events Using Actuarial Data â&#x20AC;˘ An actuary studies conditional probabilities empirically, using - accident statistics - mortality tables - insurance claims records â&#x20AC;˘ Many businesses rely on actuarial services, so a business student needs to understand the concepts of conditional probability and statistical independence. 5A-50
Independent Events Multiplication Law for Independent Events • The probability of n independent events occurring simultaneously is: P(A1 ∩ A2 ∩ ... ∩ An) = P(A1) P(A2) ... P(An) if the events are independent • To illustrate system reliability, suppose a Web site has 2 independent file servers. Each server has 99% reliability. What is the total system reliability? Let, F1 be the event that server 1 fails F2 be the event that server 2 fails 5A-51
Independent Events Multiplication Law for Independent Events • Applying the rule of independence: P(F1 ∩ F2 ) = P(F1) P(F2) = (.01)(.01) = .0001 • So, the probability that both servers are down is .0001. • The probability that at least one server is “up” is: 1 - .0001 = .9999 or 99.99% 5A-52
Independent Events Example: Space Shuttle • Redundancy can increase system reliability even when individual component reliability is low. • NASA space shuttle has three independent flight computers (triple redundancy). • Each has an unacceptable .03 chance of failure (3 failures in 100 missions). • Let Fj = event that computer j fails. 5A-53
Independent Events Example: Space Shuttle • What is the probability that all three flight computers will fail? P(all 3 fail) = P(F1 ∩ F2 ∩ F3) = P(F1) P(F2) P(F3) Õ presuming that failures are = (0.03)(0.03)(0.03) independent = 0.000027 or 27 in 1,000,000 missions. 5A-54
Independent Events The Five Nines Rule • How high must reliability be? • Public carrier-class telecommunications data links are expected to be available 99.999% of the time. • The five nines rule implies only 5 minutes of downtime per year. • This type of reliability is needed in many business situations. 5A-55
Independent Events The Five Nines Rule For example,
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Independent Events How Much Redundancy is Needed? • Suppose a certain network Web server is up only 94% of the time. What is the probability of it being down? P(down) = 1 – P(up) = 1 – .94 = .06 • How many independent servers are needed to ensure that the system is up at least 99.99% of the time (or down only 1 - .9999 = .0001 or .01% of the time)? 5A-57
Independent Events How Much Redundancy is Needed? 2 servers: P(F1 ∩ F2) = (0.06)(0.06) = 0.0036 3 servers: P(F1 ∩ F2 ∩ F3) = (0.06)(0.06)(0.06) = 0.000216 4 servers: P(F1 ∩ F2 ∩ F3 ∩ F4) = (0.06)(0.06)(0.06)(0.06) =0.00001296 • So, to achieve a 99.99% up time, 4 redundant servers will be needed. 5A-58
Applied Statistics in Business & Economics
End of Chapter 5A
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