1. Chapter 6 Lesson 6.5 Probability 6.5 Independence

2. Independent Events Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs. Two events, E and F, are said to be independent if P(E|F) = P(E). If P(E|F) = P(E), it is also true that… If two events are not independent, they are said to be dependent events.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Testing for Independences Were “Class” and “Survival” independent variables? Did one class have more or less of a chance of surviving than another class?

4. Let ’ s consider a bank that offers different types of loans: The bank offers both adjustable-rate and fixed-rate loans on single-family dwellings, condominiums and multifamily dwellings. The following table, called a joint- probability table, displays probabilities based upon the bank ’ s long-run loaning practices. Single FamilyCondoMultifamilyTotal Adjustable.40.21.09.70 Fixed.10.09.11.30 Total.50.30.20 P(Adjustable loan) =.70

5. Bank Loan ’ s Continued... Single FamilyCondoMultifamilyTotal Adjustable.40.21.09.70 Fixed.10.09.11.30 Total.50.30.20 P(Adjustable loan) =.70 P(Adjustable loan|Condo) =.21/.30 =.70 Knowing that the loan is for a condominium does not change the probability that it is an adjustable-rate loan. Therefore, the event that a randomly selected loan is adjustable and the event that a randomly selected loan is for a condo are independent.

6. *** If two events A and B are independent, then…. Multiplication Rule for Two Independent Events

7. Example 1: E 1 = event that a newly purchased monitor is not defective E 2 = event that a newly purchased mouse is not defective E 3 = event that a newly purchased disk drive is not defective E 4 = event that a newly purchased processor is not defective Suppose the four events are independent with P(E 1 ) = P(E 2 ) =.98 P(E 3 ) =.94 P(E 4 ) =.99 What is the probability that none of these components are defective? (.98)(.98)(.94)(.99) =.89 In the long run, 89% of such systems will run properly when tested shortly after purchase.

8. Let E 1 = event that a newly purchased monitor is not defective E 2 = event that a newly purchased mouse is not defective E 3 = event that a newly purchased disk drive is not defective E 4 = event that a newly purchased processor is not defective Suppose the four events are independent with P(E1) = P(E2) =.98 P(E3) =.94 P(E4) =.99 What is the probability that all these components will run properly except the monitor? (.02)(.98)(.94)(.99) =.018

9. Practice with Homework Pg.360: #6.43, 6.46, 6.47, 6.50, (*6.51), 6.54, 6.55