Probability
Dept. of AGB Veterinary College, Hebbal, Bangalore
Probability • The uncertainty of the events is numerically expressed as probability. • It measures the relative frequency of a particular event.
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Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Definition Probability refers to the chances of happening or not happening of an event. Probabilities are associated with experiments where the outcome is not known in advance or cannot be predicted. For example, if you toss a coin, will you obtain a head or tail? If you roll a die you will obtain 1, 2, 3, 4, 5 or 6?
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
DEFINITIONS 1. Event: Any phenomena occurring in nature 2. Equally likely events: All events have equal chance of occurrence 3. Mutually exclusive events: If the occurrence of an event completely avoids the occurrence of another event, then the events are said to be mutually exclusive. 4. Independent events: If the occurrence of one event in no way affects the occurrence of another event, then these events are said to be independent . Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
DEFINITIONS 5. Simple and compound event: When two or more events occur together, their happening is described as a compound event, while if only one event takes place at a time, it is called as a simple event. 6.Exhaustive cases: refer to all possible outcomes without any omissions 7. Sample space: represent the set of all possible outcomes of a phenomenon Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Measure of probability • If A is a desired event and ‘r’ is the number of occurrences of A out of the total number of possible occurrences say ‘n’, then the probability of occurrence of A, denoted by P(A), is given by: P(A )
= (No. of occurrence of A) (Total no. of occurrences)
=r/n Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Probability Probability is a ratio taking values from 0 to1. It can never be negative. If an event is an impossible event, then the probability is 0. If an event is certain to occur, then its probability is 1. Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
P (Success) = P(at least one “1”)
=1/6
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
• The value of a probability is a number between 0 and 1 inclusive. • An event that cannot occur has a probability (of happening) equal to 0 and the probability of an event that is certain to occur has a probability equal to 1
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Probability Scale
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Sample Space and Events The sample space is the set of all possible outcomes in an experiment. Example 1: If a die is rolled, the sample space S is given by S = {1,2,3,4,5,6} Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Example A die is rolled, find the probability of getting a 3.
The event of interest is "getting a 3". so E = {3}. The sample space S is given by S = {1,2,3,4,5,6}. outcomes in S is 6. Hence the probability of getting a 3 is P("3") = 1 / 6. Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Two coins are tossed, find the probability that two heads are obtained. Note: Each coin has two possible outcomes H (heads) and T (Tails). The sample space S is given by. S = {(H,T),(H,H),(T,H),(T,T)} Let E be the event "two heads are obtained". E = {(H,H)}
P(E) = n(E) / n(S) = 1 / 4 Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Example A die is rolled, find the probability of getting an even number.
The event of interest is "getting an even number". so E = {2,4,6}, the even numbers on a die. The sample space S is given by S = {1,2,3,4,5,6}. The number of possible outcomes in E is 3 and the number of possible outcomes in S is 6. Hence the probability of getting a 3 is P("3") = 3 / 6 = 1 / 2. Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Example: 3 • If two dice are rolled
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
If two dice are rolled
1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,2 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6
Sample space and events • 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) (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) }
We define an event as some specific outcome of an experiment. An event is a subsetDrof the sample space. R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
A card is drawn at random from a deck of cards. Find the probability of getting the 3 of diamond.
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Let E be the event "getting the 3 of diamond". • The sample space shows that there is one "3� of diamond" so that n(E) = 1 and n(S) = 52. Hence the probability of event E occuring is given by (E) = 1 / 52
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
A card is drawn at random from a deck of cards. Find the probability of getting a queen.
Let E be the event "getting a Queen". Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
An examination of the sample space shows that there are 4 "Queens" so that n(E) = 4 and n(S) = 52.
Hence the probability of event E occurring is given by P(E) = 4 / 52 = 1 / 13
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Mathematical probability • Mathematical or “a priori” probability is one, which can be determined deductively without any experimentation or trial.
• It is based on the assumption that one has full confidence of an event happening out of several possible alternatives (even before the event happens), which are mutually exclusive and equally likely. • It assumes that all cases are equally likely, i.e. equally probable all the time Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Statistical or empirical probability • Without assuming any of certainty of the event happening, if one tries to base the probability of an event on past experience of certain outcomes based upon a long series of experiments (that is, on the basis of statistical data), then the probability is known as statistical or empirical probability
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Statistical probability • If ‘n’ is the number of cases observed which is not only a large number but also increases indefinitely up to infinity which sets a limit to the probability of the event happening and when an event A is found to be occurring in ‘m’ number of cases, the ratio ( m / n) is close to P(A), the statistical probability. • In other words, P(A) = lim m n n Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
A jar contains 3 red marbles, 7 green marbles and 10 Yellow marbles. If a marble is drawn from the jar at random, what is the probability that this marble is Yellow? We first construct a table of frequencies that gives the marbles color distributions as follows
Color
Frequency
red
3
green Yellow
7 10
Frequency for yellow color P(E)= Total frequencies in the above table
= 10 / 20 = 1 / 2
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
The blood groups of 200 people is distributed as follows: 50 have type A blood, 65 have B blood type, 70 have O blood type and 15 have type AB blood. If a person from this group is selected at random, what is the probability that this person has O blood type? P(E)= Frequency for O blood
Group A B O AB
frequency 50 65 70 15
Total frequencies
= 70 / 200 = 0.35
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Theorems of probability 1. Addition Theorem: When two events are mutually exclusive the probability of the occurrence of either A or B is the sum of their individual probabilities. P(AUB) = P(A or B)= P(A)+P(B)
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Proof- Theorem of total probability or addition theorem Let ‘n’ be the exhaustive mutually exclusive ways in which all events can occur. Let the first event occur in a1 of these ways, the second in a2 of these ways … and kth event occur in ak of these ways. Then, if p1, p2,…, pk be the probabilities of the occurrence of these events, p1 = a1 / n1; p2 = a2 / n; p3 = a3 / n, ..., pk = ak / nk. Since these events are mutually exclusive, the number of ways in which one or other of k events will occur is: = a1+ a2+ …,+ak n = Sum of the independent probability of the events = p 1 + p2 + p 3 + pk Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Exercise If a uniform die is thrown at random, find the probability that the number facing up is a) 5 b) greater than 4 c) even number When two events are mutually exclusive the probability of the occurrence of either A or B is the sum of their individual probabilities. P(AUB) = P(A or B)= P(A)+P(B) Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Multiplication theorem of probability The probability of simultaneous occurrence of a set of independent events is the product of the separate probabilities of those independent events. If the two events A and B are independent the product of their separate probabilities gives the probability of joint occurrence. P(A ∊ B)= P(A and B)= P(A) X P(B) Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Suppose if the event A occurs in n1 ways of which m1 ways are successful and the event B occurs in n2 ways of which m2 ways are successful P(A) = m1/n1 P(B) = m2/n2 The total no. of successful outcomes for A and B is P(A) X P(B) Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Example 1: What is the probability that on two consecutive rolls of a die the numbers will be 2 and then 3?
Since the probability of getting a 2 on the first roll is 1/6 and the probability of getting a 3 on the second roll is 1/6, and since the rolls are independent of each other, simply multiply. 1/6 x 1/6 = 1/36 Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Example 2 What is the probability of tossing heads three consecutive times with a twosided fair coin?
Because each toss is independent and the odds are 1/2 for each toss, the probability is 1/2 x 1/2 x 1/2 = 1/8 Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
When two events are not mutually exclusive it is possible for both events to occur. P(AUB) = P(A or B)= P(A)+P(B) - {P(A)XP(B)}
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
Exercise From a set of chicks bearing numbers from 1 to 20, if one chick is drawn at random, find the probability that the chick will be with a number multiple of 2 or 5. When two events are not mutually exclusive it is possible for both events to occur. P(AUB) = P(A or B)= P(A)+P(B) - {P(A)XP(B)}
Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore
P(2or5)=P(2) + P(5) – {P(2)XP(5)} Number of multiples of 2 = 2,4,6,8,10,20
P(2)=10/20 =1/2 Number of multiples of 5 = 5,10,15,20
P(5) = 4/20 =1/5
1/2 +1/5 –(1/2 x 1/5) = 3/5 Dr R Jayashree, Asst. Prof, Dept of AGB Veterinary College, Hebbal, Bangalore