1. Conditional Probability

2.  A newspaper editor has 120 letters from irate readers about the firing of a high school basketball coach.  The letters are divided among parents and students, in support of or against the coach  They have space to print only one of these letters.

3. Conditional Probability  The break down of the letters:  What are the chances that a student letter supporting the coach will be chosen? Written by students Written by parents Total Support coach164460 Against coach85260 Total2496120

4. Conditional Probability  Let’s look at a Venn Diagram: Let C: event the letter is from a student Let T: event the letter favors the coach C T

5. Conditional Probability  From the Venn Diagram: Slim chance a student letter supporting the coach will be printed: Could be unfair: student letters support the coach by a ratio of 2 : 1 This fact is evident since

6. Conditional Probability  What does tell us? Given the letter came from a student, the chance it supports the coach is two- thirds In other words: 20% of the letters came from students. Of those, two-thirds were in favor of the coach

7. Conditional Probability  Notice previous Venn Diagram probabilities were all relative to sample space: For example:  looks at probability a letter supports a teacher based on a reduced sample space, student letters only

8. Conditional Probability  What does this mean? Knowing some info beforehand can change a probability  Ex: Probability of rolling a 12 with 2 dice is 1/36, but if you know the first die is a 4, the probability is 0. If the first die is a 6, the probability is 1/6 Determining a probability after some information is known is called conditional probability

9. Conditional Probability  Notation means the probability of E happening given that F has already occurred  Definition This is a conditional probability

10. Conditional Probability  The formula implies: Notice the reversal of the events E and F Note: Very Important! These are two different things. They aren’t always equal.

11. Conditional Probability  Ex: Suppose 22% of Math 115A students plan to major in accounting (A) and 67% on Math 115A students are male (M). The probability of being a male or an accounting major in Math 115A is 75%. Find and.

12. Conditional Probability  Sol: First find

13. Conditional Probability  Sol:

14. Conditional Probability  Sol:

15. Conditional Probability  Sometimes one event has no effect on another Example: flipping a coin twice  Such events are called independent events  Definition: Two events E and F are independent if or

16. Conditional Probability  Implications: So, two events E and F are independent if this is true.

17. Conditional Probability  The property of independence can be extended to more than two events: assuming that are all independent.

18. Conditional Probabilities  INDEPENDENT EVENTS AND MUTUALLY EXCLUSIVE EVENTS ARE NOT THE SAME Mutually exclusive: Independence:

19. Conditional Probability  Ex: Suppose we roll toss a fair coin 4 times. Let A be the event that the first toss is heads and let B be the event that there are exactly three heads. Are events A and B independent?

20. Conditional Probability  Soln: For A and B to be independent, and Different, so dependent

21. Conditional Probability  Ex: Suppose you apply to two graduate schools: University of Arizona and Stanford University. Let A be the event that you are accepted at Arizona and S be the event of being accepted at Stanford. If and, and your acceptance at the schools is independent, find the probability of being accepted at either school.

22. Conditional Probability  Soln: Find. Since A and S are independent,

23. Conditional Probability  Soln: There is a 76% chance of being accepted by a graduate school.

24. Conditional Probability  Independence holds for complements as well.  Ex: Using previous example, find the probability of being accepted by Arizona and not by Stanford.

25. Conditional Probability  Soln: Find.

26. Conditional Probability  Ex: Using previous example, find the probability of being accepted by exactly one school.  Sol: Find probability of Arizona and not Stanford or Stanford and not Arizona.

27. Conditional Probability  Sol: (continued) Since Arizona and Stanford are mutually exclusive (you can’t attend both universities) (using independence)

28. Conditional Probability  Soln: (continued)

29. Conditional Probability  Independence holds across conditional probabilities as well.  If E, F, and G are three events with E and F independent, then

30. Conditional Probability  Focus on the Project: Recall: and However, this is for a general borrower Want to find probability of success for our borrower

31. Conditional Probability  Focus on the Project: Start by finding and We can find expected value of a loan work out for a borrower with 7 years of experience.

32. Conditional Probability  Focus on the Project: To find we use the info from the DCOUNT function This can be approximated by counting the number of successful 7 year records divided by total number of 7 year records

33. Conditional Probability  Focus on the Project: Technically, we have the following: So, Why “technically”? Because we’re assuming that the loan workouts BR bank made were made for similar types of borrowers for the other three. So we’re extrapolating a probability from one bank and using it for all the banks.

34. Conditional Probability  Focus on the Project: Similarly, This can be approximated by counting the number of failed 7 year records divided by total number of 7 year records

35. Conditional Probability  Focus on the Project: Technically, we have the following: So,

36. Conditional Probability  Focus on the Project: Let be the variable giving the value of a loan work out for a borrower with 7 years experience Find

37. Conditional Probability  Focus on the Project: This indicates that looking at only the years of experience, we should foreclose (guaranteed $2.1 million)

38. Conditional Probability  Focus on the Project: Of course, we haven’t accounted for the other two factors (education and economy) Using similar calculations, find the following:

39. Conditional Probability  Focus on the Project:

40. Conditional Probability  Focus on the Project: Let represent value of a loan work out for a borrower with a Bachelor’s Degree Let represent value of a loan work out for a borrower with a loan during a Normal economy

41. Conditional Probability  Focus on the Project: Find and

42. Conditional Probability  Focus on the Project:  So, two of the three individual expected values indicates a foreclosure:

43. Conditional Probability  Focus on the Project: Can’t use these expected values for the final decision None has all 3 characteristics combined: for example has all education levels and all economic conditions included

44. Conditional Probability  Focus on the Project: Now perform some calculations to be used later We will use the given bank data: That is is really and so on…

45. Conditional Probability  Focus on the Project: We can find since Y, T, and C are independent Also

46. Conditional Probability  Focus on the Project: Similarly:

47. Conditional Probability  Focus on the Project:

48. Conditional Probability  Focus on the Project:

49. Conditional Probability  Focus on the Project:

50. Conditional Probability  Focus on the Project: Now that we have found and we will use these values to find and