Law of Total Probability and Bayes’ Rule

1. Law of Total Probabilityand Bayes’ Rule

2. “Event-composition method” • Understand the experiment and sample points. • Using set notation, express the event of interest in terms of events for which the probability is known. • Applying probability rules, combine the known probabilities to determine the probability of the specified event.

3. Problem 2.86 • In a factory, 40% of items produced come from Line 1 and others from Line 2. • Line 1 has a defect rate of 8%. Line 2 has a defect rate of 10%. • For randomly selected item, find probability the item is not defective.A: the selected item is not defective

4. B1 B2 Problem 2.86 • A: the selected item is not defective. S A • B1: item came from Line 1.B2: item came from Line 2.

5. Or, in terms of conditional probabilities Problem 2.86 • So we may write • Since this is the union of disjoint sets,the Additive Law yields

6. The Decision Tree defective Line 1 not defective defective Line 2 not defective

7. Problem 2.94 • Must find blood donor for an accident victim in the next 8 minutes or else… • Checking blood types of potential donors requires 2 minutes each and may only be tested one at a time. • 40% of the potential donors have the required blood type. • What is the probability a satisfactory blood donor is identified in time to save the victim?

8. 4 mutually exclusive events Finding a Donor • A: blood donor is found within 8 minutes • Some sample points: “B bad, G good”A = { (G), (B,G), (B,B,G), (B,B,B,G) } • Let Ai: ith donor has correct blood type

9. Finding a Donor • Trials are independent andeach P(Ai) = 0.40,and so

10. Finding a Donor saved! saved! saved! saved! too late!

11. Problem 2.96 • Of 6 refrigerators, 2 don’t work. • The refrigerators are tested one at a time. • When tested, it’s clear whether it works! • What is the probability the last defective refrigerator is found on the 4th test? • What is the probability no more than 4 need to be tested to identify both defective refrigerators?

12. Problem 2.96 • Given that exactly one defective refrigerator was found during the first 2 tests, what is the probability the other one is found on the 3rd or 4th test?

13. Partition of the Sample Space S … B1 B2 Bk

14. Union of Disjoint Sets S A … B1 B2 Bk

15. B1 B2 Recall Problem 2.86 • A: the selected item is not defective. S A Not defective • B1: item came from Line 1.B2: item came from Line 2.

16. Law of Total Probability S A … B1 B2 Bk

17. Total Probability P(A|B1)P(B1) B1 P(A|B2)P(B2) B2 B3 P(A|B3)P(B3)

18. Bayes’ Theorem follows…

19. P(A|B1)P(B1) B1 P(A|B2)P(B2) B2 B3 P(A|B3)P(B3) Bayes’

20. Making Resistors • Three machines M1, M2, and M3 produce “1000-ohm” resistors. • M1 produces 80% of resistors accurate to within 50 ohms, M2 produces 90% to within 50 ohms, and M3 produces 60% to within 50 ohms. • Each hour, M1 produces 3000 resistors, M2 produces 4000, and M3 produces 3000. • If all of the resistors are mixed together and shipped in a single container, what is the probability a selected resistor is accurate to within 50 ohms?

21. Making Resistors • Define A: resistor is accurate to within 50 ohms. • M1 produces 80% of resistors accurate to within 50 ohms, M2 produces 90% to within 50 ohms, and M3 produces 60% to within 50 ohms. • Each hour, M1 produces 3000 resistors, M2 produces 4000, and M3 produces 3000.

22. Using Total Probability That is, 78 % are expected to be accurate to within 50 ohms.

23. The Tree (0.8)(0.3) M1 (0.9)(0.4) M2 M3 (0.6)(0.3)

24. …given it’s within 50 ohms… • Determine the probability that,given a selected resistor is accurate to within 50 ohms, it was produced by M1. P( M1 | A) = ? • Determine the probability that,given a selected resistor is accurate to within 50 ohms, it was produced by M3. P( M3 | A) = ?

25. Given A… (0.8)(0.3) M1 (0.9)(0.4) M2 M3 (0.6)(0.3)

26. Arthritis • A test detects a particular type of arthritis for individuals over 50 years old. • 10% of this age group suffers from this arthritis. • For individuals in this age group known to have the arthritis, the test is correct 85% of the time. • For individuals in this age group known to NOT have the arthritis, the test indicates arthritis (incorrectly!) 4% of the time. • P( has arthritis | tests positive ) = ?

27. Arthritis • 10% of this age group suffers from this arthritis.P(have arthritis) = 0.10 • For individuals in this age group known to have the arthritis, the test is correct 85% of the time.P( tests positive | have arthritis ) = 0.85 • For individuals in this age group known to NOT have the arthritis, the test indicates arthritis (incorrectly!) 4% of the time.P(tests positive | no arthritis ) = 0.04 • P( have arthritis | tests positive ) = ?

28. 0.85 positive Hasarthritis negative 0.1 0.9 Noarthritis 0.04 positive negative

29. The 3 Urns • Three urns contain colored balls. Urn Red White Blue 1 3 4 1 2 1 2 3 3 4 3 2 • An urn is selected at random and one ball is randomly selected from the urn. • Given that the ball is red, what is the probability it came from urn #2 ?

30. The Lost Labels • A large stockpile of cases of light bulbs, 100 bulbs to a box, have lost their labels. • The boxes of bulbs come in 3 levels of quality: high, medium, and low. • It’s known 50% of the boxes were high quality, 25% medium, and 25% low. • Two bulbs will be tested from a box to check if they’re defective.

31. Lost Labels… • The likelihood of finding defective bulbs is dependent on the bulb quality: Number of defects Low Medium High 0 .49 .64 .81 1 .42 .32 .18 2 .09 .04 .01 • Given neither bulb is found to be defective, what is the probability the bulbs came from a box of high quality bulbs?