Conditional Probability

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2 Test 1 Results You will get back your test 1 papers on Friday. There is a generous nonlinear curve of the scores: if x is your raw score, your grade out of 100 can be computed as 100x x x = = Arthur Berg 2/ 11

3 Grade Distributions, Old and New Arthur Berg 3/ 11

4 Unemployment Rate (job-seekers) (job-seekers + employed) = 6.1% Arthur Berg 4/ 11

5 Unemployment by Education Arthur Berg 5/ 11

6 Relative Frequencies of Employment by Education Arthur Berg 6/ 11

7 Example 3.1 Conditional distribution on Men and conditional distribution on Women: Arthur Berg 7/ 11

8 Probability of Two Girls 1 What is the probability that a family with two children has two girls? 2 If you know the given family has at least one girl, what is the probability of the family having two girls? Definition If A and B are any two events, then the conditional probability of A given B, denoted by P(A B), is P(A B) = P(AB) P(B) provided that P(B) > 0. P(AB) can then be computed in two ways as: P(AB) = P(A B)P(B) P(AB) = P(B A)P(A) Arthur Berg 8/ 11

9 Probability of Two Girls 1 What is the probability that a family with two children has two girls? S = {BB, BG, GB, GG} where each is equally likely, so the probability is 1/4. 2 If you know the given family has at least one girl, what is the probability of the family having two girls? Definition If A and B are any two events, then the conditional probability of A given B, denoted by P(A B), is P(A B) = P(AB) P(B) provided that P(B) > 0. P(AB) can then be computed in two ways as: P(AB) = P(A B)P(B) P(AB) = P(B A)P(A) Arthur Berg 8/ 11

10 Probability of Two Girls 1 What is the probability that a family with two children has two girls? S = {BB, BG, GB, GG} where each is equally likely, so the probability is 1/4. 2 If you know the given family has at least one girl, what is the probability of the family having two girls? Definition If A and B are any two events, then the conditional probability of A given B, denoted by P(A B), is P(A B) = P(AB) P(B) provided that P(B) > 0. P(AB) can then be computed in two ways as: P(AB) = P(A B)P(B) P(AB) = P(B A)P(A) Arthur Berg 8/ 11

11 Probability of Two Girls 1 What is the probability that a family with two children has two girls? S = {BB, BG, GB, GG} where each is equally likely, so the probability is 1/4. 2 If you know the given family has at least one girl, what is the probability of the family having two girls? S = {BG, GB, GG} where each is equally likely, so the probability is 1/3. Definition If A and B are any two events, then the conditional probability of A given B, denoted by P(A B), is P(A B) = P(AB) P(B) provided that P(B) > 0. P(AB) can then be computed in two ways as: P(AB) = P(A B)P(B) P(AB) = P(B A)P(A) Arthur Berg 8/ 11

12 Probability of Two Girls 1 What is the probability that a family with two children has two girls? S = {BB, BG, GB, GG} where each is equally likely, so the probability is 1/4. 2 If you know the given family has at least one girl, what is the probability of the family having two girls? S = {BG, GB, GG} where each is equally likely, so the probability is 1/3. Definition If A and B are any two events, then the conditional probability of A given B, denoted by P(A B), is P(A B) = P(AB) P(B) provided that P(B) > 0. P(AB) can then be computed in two ways as: P(AB) = P(A B)P(B) P(AB) = P(B A)P(A) Arthur Berg 8/ 11

13 Example 3.2 Example (Example 3.2, p. 64) There are four batteries, and one is defective. Two are to be selected at random for use on a particular day. Find the probability that the second battery selected is not defective, given that the first was not defective. Let N i denote the event that the i th battery selected is nondefective. P(N 2 N 1 ) = P(N 1N 2 ) P(N 1 ) where and P(N 1 N 2 ) = = 1 2 P(N 1 ) = 3 4 Therefore P(N 2 N 1 ) = 2 3. Arthur Berg 9/ 11

14 Example 3.2 Arthur Berg 10/ 11

15 Three axioms of probability Conditional probabilities satisfy the three axioms of probability. (i) 0 P(A B) 1 (ii) P(S B) = 1 (iii) If A 1, A 2,... are mutually exclusive events, then so are A 1 B, A 2 B,..., and ( ) P A i B = P(A i B) i=1 i=1 Arthur Berg 11/ 11

Chapter 3 Conditional Probability and Independence. Wen-Guey Tzeng Computer Science Department National Chiao Tung University

Chapter 3 Conditional Probability and Independence Wen-Guey Tzeng Computer Science Department National Chiao Tung University Conditional probability P(A B) = the probability of event A given the occurrence