Bayes’ Theorem

1. Bayes’ Theorem

2. Bayes’ Theorem • An insurance company divides its clients into two categories: those who are accident prone and those who are not. Statistics show there is a 40% chance an accident prone person will have an accident within 1 year whereas there is a 20% chance non-accident prone people will have an accident within the first year. • If 30% of the population is accident prone, what is the probability that a new policyholder has an accident within 1 year?

3. Bayes’ Theorem • Let A be the event a person is accident prone • Let F be the event a person has an accident within 1 year A AC F

4. Bayes’ Theorem • Notice we’ve divided up or partitioned the sample space along accident prone and non-accident prone A AC F

5. Bayes’ Theorem • Notice that and are mutually exclusive events and that • Therefore • We need to find and • How?

6. Bayes’ Theorem • Recall from conditional probability

7. Bayes’ Theorem • Thus: P(A) = 0.30 since 30% of population is accident prone P(F|A) = 0.40 since if a person is accident prone, then his chance of having an accident within 1 year is 40% P(F|AC) = 0.2 since non-accident prone people have a 20% chance of having an accident within 1 year P(AC) = 1- P(A) = 0.70

8. Bayes’ Theorem • Updating our Venn Diagram • Notice again that A AC F

9. Bayes’ Theorem • So the probability of having an accident within 1 year is:

10. Bayes’ Theorem • Using Tree Diagrams: Accident w/in 1 year P(F|A)=0.40 Accident Prone P(A) = 0.30 No Accident w/in 1 year P(FC|A)=0.60 Accident w/in 1 year P(F|AC)=0.20 Not Accident Prone P(AC) = 0.70 No Accident w/in 1 year P(FC|AC)=0.80

11. Bayes’ Theorem • Notice you can have an accident within 1 year by following branch A until F is reached • The probability that F is reached via branch A is given by • In other words, the probability of being accident prone and having one within 1 year is

12. Bayes’ Theorem • You can also have an accident within 1 year by following branch ACuntil F is reached • The probability that F is reached via branch ACis given by • In other words, the probability of NOT being accident prone and having one within 1 year is

13. Bayes’ Theorem • What would happen if we had partitioned our sample space over more events, say , all them mutually exclusive? • Venn Diagram A1 A2 An-1 An . . . . . . (etc.) F

14. Bayes’ Theorem • For each

15. Bayes’ Theorem • Tree Diagram

16. Bayes’ Theorem • Notice that F can be reached via branches • Multiplying across each branch tells us the probability of the intersection • Adding up all these products gives:

17. Bayes’ Theorem • Ex: 2 (text tractor example) Suppose there are 3 assembly lines: Red, White, and Blue. Chances of a tractor not starting for each line are 6%, 11%, and 8%. We know 48% are red and 31% are blue. The rest are white. What % don’t start?

18. Bayes’ Theorem • Soln. R: red P(R) = 0.48 W: white P(W) = 0.21 B: blue P(B) = 0.31 N: not starting P(N | R) = 0.06 P(N | W) = 0.11 P(N | B) = 0.08

19. Bayes’ Theorem • Soln.

20. Bayes’ Theorem • Main theorem: Suppose we know . We would like to use this information to find if possible. Discovered by Reverend Thomas Bayes

21. Bayes’ Theorem • Main theorem: • Ex. Suppose and partition a space and A is some event. • Use and to determine .

22. Bayes’ Theorem • Recall the formulas: • So,

23. Bayes’ Theorem • Bayes’ Theorem:

24. Bayes’ Theorem • Ex. 4 (text tractor example) 3 assembly lines: Red, White, and Blue. Some tractors don’t start (see Ex. 2). Find prob. of each line producing a non-starting tractor. P(R) = 0.48 P(N | R) = 0.06 P(W) = 0.21 P(N | W) = 0.11 P(B) = 0.31 P(N | B) = 0.08

25. Bayes’ Theorem • Soln. Find P(R | N), P(W | N), and P(B | N) P(R) = 0.48 P(N | R) = 0.06 P(W) = 0.21 P(N | W) = 0.11 P(B) = 0.31 P(N | B) = 0.08

26. Bayes’ Theorem • Soln.

27. Bayes’ Theorem • Soln.

28. Bayes’ Theorem • Focus on the Project: We want to find the following probabilities: and . To get these, use Bayes’ Theorem

29. Bayes’ Theorem • Focus on the Project:

30. Bayes’ Theorem • Focus on the Project: In Excel,we find the probability to be approx. 0.4774

31. Bayes’ Theorem • Focus on the Project: In Excel,we find the probability to be approx. 0.5226

32. Bayes’ Theorem • Focus on the Project: Let Z be the value of a loan work out for a borrower with 7 years, Bachelor’s, Normal…

33. Bayes’ Theorem • Focus on the Project: Since foreclosure value is $2,100,000 and on average we would receive $2,040,000 from a borrower with John Sanders characteristics, we should foreclose.

34. Bayes’ Theorem • Focus on the Project: However, there were only 239 records containing 7 years experience. Look at range of value 6, 7, and 8 (1 year more and less)

35. Bayes’ Theorem • Focus on the Project: Use DCOUNT function with an extra “Years in Business” heading Same for “no” Added a new column

36. Bayes’ Theorem • Focus on the Project: From this you get 349 successful and 323 failed records Let be a borrower with 6, 7, or 8 years experience and

37. Bayes’ Theorem • Focus on the Project:

38. Bayes’ Theorem • Focus on the Project: Use Bayes’ Theorem to get new probabilities : 6, 7, or 8 years, Bachelor’s, Normal (indicates work out)

39. Bayes’ Theorem • Focus on the Project: We can look at a large range of years. Look at range of value 5, 6, 7, 8, and 9 (2 years more and less)

40. Bayes’ Theorem • Focus on the Project: Use DCOUNT function with an extra “Years in Business” heading Same for “no” Added a new column

41. Bayes’ Theorem • Focus on the Project: From this you get 566 successful and 564 failed records Let be a borrower with 5, 6, 7, 8, or 9 years exper. and

42. Bayes’ Theorem • Focus on the Project:

43. Bayes’ Theorem • Focus on the Project: Use Bayes’ Theorem to get new probabilities : 5, 6, 7, 8, or 9 years, Bachelor’s, Normal (indicates work out)

44. Bayes’ Theorem • Focus on the Project: Since both indicated a work out while only indicated a foreclosure, we will work out a new payment schedule. Any further extensions…