Intersection of Events and the Multiplication Rule

1. Intersection of Events and the Multiplication Rule Section 4.8

2. Intersection of Events • The outcomes that are common to both events. • Denoted by “A and B” or “A ∩ B”

3. Multiplication Rule for Dependent Events • To find the probability of two or more events happening together. • Joint Probability- P(A and B), is found by multiplying the marginal probability of one event by the conditional probability of the second event. • P(A and B) = P(A) P(B | A) • Joint Probability of mutually exclusive events is zero.

4. Example: Find the probability that the employee is female and a graduate. P(F ∩ G) = P(F) P(G | F)

5. Example: Find the probability that the employee is female and a graduate. P(F ∩ G) = P(F) P(G | F)

6. Example: Find the probability that the employee is female and a graduate. P(F ∩ G) = P(F) P(G | F)

7. Four out of 20 CD’s are defective, find P(2 CD’s are defective) if selected without replacement.

8. Calculating Conditional Probability • P(B | A) = P(A ∩ B) / P(A) • Example: P(student is a senior) = .2 and P(student is a senior and a science major) = .03. Find P(student is a science major | senior) .03/.2 = .15

9. Multiplication Rule for Independent Events • P(A ∩ B) = P(A) P(B) • Example: P(allergic to penicillin) = .2 Find P(3 patients are allergic) = .008 Find P(atleast one is not allergic) = complement of “all three are allergic” = 1-.008 = .992

10. Tree Diagram of Penicillin Example