Section 4.1 (cont.) Probability Trees

1. Section 4.1 (cont.) Probability Trees A Graphical Method for Complicated Probability Problems

2. Example: Southwest Energy • A Southwest Energy Company pipeline has 3 safety shutoff valves in case the line starts to leak. • The valves are designed to operate independently of one another: • 7% chance that valve 1 will fail • 10% chance that valve 2 will fail • 5% chance that valve 3 will fail • If there is a leak in the line, find the following probabilities: • That all three valves operate correctly • That all three valves fail • That only one valve operates correctly • That at least one valve operates correctly

3. A: P(all three valves operate correctly) P(all three valves work) = .93*.90*.95 = .79515

4. B: P(all three valves fail) P(all three valves fail) = .07*.10*.05 = .00035

5. C: P(only one valve operates correctly) P(only one valve operates correctly = P(only V1 works) +P(only V2 works) +P(only V3 works) = .93*.10*.05 +.07*.90*.05 +.07*.10*.95 = .01445

6. D: P(at least one valve operates correctly) 7 paths P(at least one valve operates correctly = 1 – P(no valves operate correctly) = 1 - .00035 = .99965 1 path

7. Example: AIDS Testing • V={person has HIV}; CDC: Pr(V)=.006 • P : test outcome is positive (test indicates HIV present) • N : test outcome is negative • clinical reliabilities for a new HIV test: • If a person has the virus, the test result will be positive with probability .999 • If a person does not have the virus, the test result will be negative with probability .990

8. Question 1 • What is the probability that a randomly selected person will test positive?

9. Probability Tree Approach • A probability tree is a useful way to visualize this problem and to find the desired probability.

10. Probability Tree Multiply branch probs clinical reliability clinical reliability

11. Question 1 Answer • What is the probability that a randomly selected person will test positive? • Pr(P )= .00599 + .00994 = .01593

12. Question 2 • If your test comes back positive, what is the probability that you have HIV? (Remember: we know that if a person has the virus, the test result will be positive with probability .999; if a person does not have the virus, the test result will be negative with probability .990). • Looks very reliable

13. Question 2 Answer Answer two sequences of branches lead to positive test; only 1 sequence represented people who have HIV. Pr(person has HIV given that test is positive) =.00599/(.00599+.00994) = .376

14. Summary • Question 1: • Pr(P ) = .00599 + .00994 = .01593 • Question 2: two sequences of branches lead to positive test; only 1 sequence represented people who have HIV. Pr(person has HIV given that test is positive) =.00599/(.00599+.00994) = .376

15. Recap • We have a test with very high clinical reliabilities: • If a person has the virus, the test result will be positive with probability .999 • If a person does not have the virus, the test result will be negative with probability .990 • But we have extremely poor performance when the test is positive: Pr(person has HIV given that test is positive) =.376 • In other words, 62.4% of the positives are false positives! Why? • When the characteristic the test is looking for is rare, most positives will be false.

16. examples 1. P(A)=.3, P(B)=.4; if A and B are mutually exclusive events, then P(AB)=? A B = , P(A B) = 0 2. 15 entries in pie baking contest at state fair. Judge must determine 1st, 2nd, 3rd place winners. How many ways can judge make the awards? 15P3 = 2730