Mathematical Statistics İST2011 PROBABILITY THEORY (3) DEU, DEPARTMENT OF STATISTICS MATHEMATICAL STATISTICS SUMMER SEMESTER, 2017.

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1 Mathematical Statistics İST2011 PROBABILITY THEORY (3) 1 DEU, DEPARTMENT OF STATISTICS MATHEMATICAL STATISTICS SUMMER SEMESTER, 2017 If the five balls are places in five cell at random, find the probability p that exactly one cell is empty? 1

2 Two cards are selected at random from 10 cards numbered 1 to 10. Find the probability p that the sum is odd if I. The two cards are drawn together II. The two cards are drawn one after the other without replacement III. The two cards are drawn one after the other with replacement Conditional Probability 4 A major objective of probability modelling is to determine how likely it is that an event A will occur when a certain experiment is performed. However, in numerous cases the probability assigned to A will be affected by knowledge of the occurence of another event B. 2

3 Conditional Probability 5 In such an example, we will use the terminology conditional probability of A given B and the notation P(A B) will be used to disttinguish between this new concept and ordinary probability P(A). 6 A consumer research organization has studied the services under warranty provided by the 50 new-car dealers in a certain city, and its findings are summarized in the following table: Good Service under warranty Poor Service under warranty In business 10 years or more In business less than 10 years

4 (Cont.) 7 a) If a person randomly selects one of these new-car dealers, what is the probability that he gets one who provides good service under warranty? b) Also, if a person randomly selects one of the dealers who has been in business for 10 years or more, what is the probability that he gets one who provides good service under warranty? (Cont.) G: the selection of a dealer who provides good service under warranty T:the selection of a dealer who has been in business 10 years or more 8 Since the numerator of P(G T) is n(t G 16 in the preceeding example, the number of dealers who have been in business for 10 years or more and provide good service under warranty, and the denominator is n T, the number of dealers who have been in business for 10 years or more. 4

5 (Cont.) 9 Symbolically, Then if we divide the numerator and the denominator by n(s), the total number of new-car dealers in the given city, we get and we have, thus expressed the conditional probability P(G T) in terms of two probability defined for the whole sample space S. Definition: Conditional Probability If A and B are any two events in a sample space S and P(B) 0, the conditional probability of A given B is 10 5

6 Properties of Conditional Probability If P B 0, we have 1. P A B 0 2. P B B 1 3. If are mutually exclusive events, then From probability axioms, we have Properties of Conditional Probability a. P(A B)>P(A) P B A P B P A B P A P B positive correlation b. P(A B)<P(A) P B A P B P A B P A P B negative correlation c. P(A B)=P(A) P(B A)=P(B) P(A B)=P(A)P(B) independence 6

7 Theorem: Multiplication Rule 13 For any events A and B, P(A B)=P(A B)P(B)=P(B A)P(A). This procedure would be extended to three or more events occur: P(A B C)=P(C A B)P(B A)P(A) 14 An urn contains five white chips, four black chips and three red chips. Four chips are drawn sequentially and without replacement. What is the probability of obtaining the sequence (W,R,W,B)? 7

8 Total Probability and Bayes Rule Let A and B be events. We may express A as 15 or more generally, If are mutually exclusive and exhaustive in the sense that then Theorem (Law of Total Probability) 16 If is a collection of mutually exclusive and exhaustive events, then for any event A, Proof: 8

9 17 An insurance company believes that people can be divided into two classes: those who are accident prone and those are not. Their statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a non-accident prone person. If we assume that 30 percent of the population is accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing policy? Bayes Rule 18 If is a collection of mutually exclusive and exhaustive events, then for each j = 1,..,k Proof: 9

10 19 Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? 20 A lie-detector test will give apositive result 99 percent of the time when a person is actually lying, but it will also give a positive (false) result 1 percent of the time when a person is telling the truth. It is estimated that 10 percent of the people who take the test are lying. What is the probability that aperson chosen at random will test positively? 10

11 Independent Events 21 Informally speaking, two events A and B are independent if the occurrence of nonoccurrence of either one does not affect the prob. of the occurrence of the other. Symbolically, two events A and B are independent if P(B A)=P(B) and P(A B)=P(A), and it can be shown that either of these equalities implies the other when both of the conditional prob. exist, when neither P(A) nor P(B) equals zero. Definition (Independent Events) Two events A and B are called independent events if 22 P(A B) P(A)P(B) Otherwise, A and B are called dependent events. 11

12 Note: If A and B are mutually exclusive then, 23 P(A B) P(BA) 0 whereas for independent nonnull events the conditional probabilities are nonzero. Series System Consider a series system of two components C 1 and C 2. Let A 1 ={C 1 fails} A 2 ={C 2 fails} F={the system fails} is A1 A 2 24 P(A1 A2) P(A 1) P(A2) P(A1 A2) Suppose that P(A 1 )=0.1 and P(A 2 )=0.2 If we assume that A 1 and A 2 are independent, then the prob. that system fails is P 2 (F) P(A1 A ) The prob. that the system works properly is =

13 Parallel System 25 For a parallel system to fail, it is necessary that both components fail, so the event the system fails is The prob. That this system fails is A 1 A 2 P(A1 A2) P(A 1) P(A2) P 2 (F) P(A1 A ) Again assuming the components fail independently. Series&Parallel System 26 C 1 C 2 Series System C 1 C 2 Parallel System 13

14 Theorem If A and B are independent, then AandB C 2. A C and B 3. A C and B C are also independent. Proof: 28 A company employs 2 managers. The prob. that A is unable to work due to illness is 0.04 and the prob. that B is unable to work due to illness is Assuming that the health status of the two managers are independent, calculate the prob. that both are able to work? P(A C C B )? Solt: 14

15 Definition:Mutually Independence 29 The k events A 1,A 2,,A k are said to be independent or mutually independent if for every j=2,3,,k and every subset of distinct indices i 1,i 2,,i j. P(A i1 A i2... A ) P(A ij i1 )P(A i2 )...P(A ij ) 30 Lets consider an experiment in which a computer randomly selects an integer between 1 and 9. (inclusive) S={1,2,,9} Let A={1,2,3}, B={2,5,8} and C={2,4,6} Prove that the events A, B, and C are pairwise independent and the events A, B, and C are not mutually independent. Proof: 15

16 Product Set 31 Suppose first that we have n sets, S 1, S 2,, S n. The Cartesian product (named for René Descartes) of S 1,S 2,,S n denoted S 1 xs 2 x xs n is the set of all (ordered) sequences (S 1,S 2,,S n )where S i is an element of S i for each i. Product Set 32 If we have n experiments with sample spaces S 1,S 2,,S n,thens 1 xs 2 x x S n is the natural sample space for the compound experiment that consists of performing the n experiments in sequence if S i =S for each i, then the product set can be written conpactly as S n =S 1 xs 2 x xs n (n factors) 16

17 Product Set/Exp 33 A coin-die experiment, we have a coin and a die. First the coin is tossed, and then the die is rolled. The coin score and the score of die are recorded. Let A denote the event that the die score is at least 4. Let B denote the event that the coin score is head. Define the sample space S mathematically. ExpressAasasubsetofS. Is A and B independent? Product Set/Exp 34 In a certain district, candidates 1,2 and 3 are running for congress. A political consulting samples 1000 registered voters from the district and records the age, gender and candidate preference of each person in the sample. Asuume that a registered voter must be at least 18 years old. Define a sample space for experiment. 17