1. 4-4 Multiplication Rules and Conditional Probability Objectives -understand the difference between independent and dependent events -know how to use multiplication rule to calculate probability of independent events. -know how to use conditional probability to calculate probability of dependent events.

2. independentTwo events A and B are independent if the fact that A occurs does not affect the probability of B occurring. Example:Example: Rolling a die and getting a 6, and then rolling another die and getting a 3 are independent events. 4-4 The Multiplication Rules and Conditional Probability

3. 4-4 Multiplication Rules

4. 4-4 Multiplication Rule 1 - 4-4 Multiplication Rule 1 - Example A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen and then an ace. Solution:Solution: Because these two events are independent (why?), P(queen and ace) = (4/52)  (4/52) = 16/2704 = 1/169.

5. 4-4 Multiplication Rule 1 - 4-4 Multiplication Rule 1 - Example A Decima poll found that 46% of Canadians say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say that they suffer stress at least once a week. Solution:Solution: Let S denote stress. Then P(S and S and S) = (0.46) 3 = 0.097.

6. 4-4 Multiplication Rule 1 - 4-4 Multiplication Rule 1 - Example The probability that a specific medical test will show positive is 0.32. If four people are tested, find the probability that all four will show positive. Solution:Solution: Let T denote a positive test result. Then P(T and T and T and T) = (0.32) 4 = 0.010.

7. When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent. Example:Example: Having high grades and getting a scholarship are dependent events. Multiplication Rules: Conditional Probability

8. conditional probabilityThe conditional probability of an event B in relationship to an event A is the probability that an event B occurs after event A has already occurred. The notation for the conditional probability of B given A is P(B|A). NOTE:NOTE: This does not mean B  A. Multiplication Rules: Conditional Probability

9. 4-4 Multiplication Rule 2

10. In a shipment of 25 microwave ovens, two are defective. If two ovens are randomly selected and tested, find the probability that both are defective if the first one is not replaced after it has been tested. Solution:Solution: See next slide. The Multiplication Rules: Conditional Probability - The Multiplication Rules: Conditional Probability - Example

11. Solution:Solution: Since the events are dependent, P(D 1 and D 2 ) = P(D 1 )  P(D 2 | D 1 ) = (2/25)(1/24) = 2/600 = 1/300. The Multiplication Rules and Conditional Probability - The Multiplication Rules and Conditional Probability - Example

12. The KW Insurance Company found that 53% of the residents of a city had homeowner’s insurance with its company. Of these clients, 27% also had automobile insurance with the company. If a resident is selected at random, find the probability that the resident has both homeowner’s and automobile insurance. The Multiplication Rules and Conditional Probability - The Multiplication Rules and Conditional Probability - Example

13. Solution:Solution: Since the events are dependent, P(H and A) = P(H)  P(A|H) = (0.53)(0.27) = 0.1431. The Multiplication Rules and Conditional Probability - The Multiplication Rules and Conditional Probability - Example

14. Box 1 contains two red marbles and one blue marble. Box 2 contains three blue marbles and one red marble. A coin is tossed. If it falls heads up, box 1 is selected and a marble is drawn. If it falls tails up, box 2 is selected and a marble is drawn. Find the probability of selecting a red marble. The Multiplication Rules and Conditional Probability - The Multiplication Rules and Conditional Probability - Example

15. Tree Diagram for Tree Diagram for Example P(B 1 ) 1/2 Red Blue Box 1 P(B 2 ) 1/2 Box 2 P(R|B 1 ) 2/3 P(B|B 1 ) 1/3 P(R|B 2 ) 1/4 P(B|B 2 ) 3/4 (1/2)(2/3) (1/2)(1/3) (1/2)(1/4) (1/2)(3/4)

16. Solution:Solution: P(red) = (1/2)(2/3) + (1/2)(1/4) = 2/6 + 1/8 = 8/24 + 3/24 = 11/24. The Multiplication Rules and Conditional Probability - The Multiplication Rules and Conditional Probability - Example

17. Conditional Probability - Formula

18. The probability that Sam parks in a no- parking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in the no- parking zone is 0.2. On Tuesday, Sam arrives at school and has to park in a no- parking zone. Find the probability that he will get a ticket. Conditional Probability - Conditional Probability - Example

19. Solution:Solution: Let N = parking in a no- parking zone and T = getting a ticket. Then P(T |N) = [P(N and T) ]/P(N) = 0.06/0.2 = 0.30. Conditional Probability - Conditional Probability - Example

20. A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results are shown in the table on the next slide. Conditional Probability - Conditional Probability - Example

22. Find the probability that the respondent answered “yes” given that the respondent was a female. Solution:Solution: Let M = respondent was a male; F = respondent was a female; Y = respondent answered “yes”; N = respondent answered “no”. Conditional Probability - Conditional Probability - Example

23. P(Y|F) = [P( F and Y) ]/P(F) = [8/100]/[50/100] = 4/25. Find the probability that the respondent was a male, given that the respondent answered “no”. Solution: P(M|N) = [P(N and M)]/P(N) = [18/100]/[60/100] = 3/10. Conditional Probability - Conditional Probability - Example

24. Homework Pg. 215 # 1-49 every other odd