Conditional Probability

1. Conditional Probability

2. Conditional Probability • 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?

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 This is a conditional probability • Notation • means the probability of E happening given that F has already occurred • Definition

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

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