Probability and Decision Making

1. Probability and Decision Making

2. Many problems with decision making can be made less stressful with an understanding of probability. But , almost nothing can be more confusing, and create more errors than an incorrect application of probability.

3. The human mind is notoriously bad at estimating probabilities….

5. In a survey of faculty at the University of Nebraska, 68% rated themselves in the top 25% for teaching ability. In a similar survey, 87% of MBA students at Stanford University rated their academic performance as above the median. For driving skill, 93% of the US sample put themselves in the top 50% (above the median).

6. Chance of winning the lottery with 6 correct numbers (1-49) is 1 in 13,983,816 Chance of dying in a car accident is 1.49 in 100 A person is 21,000 times more likely to die in a car than win the lottery!

7. Sample Space: All possible outcomes The sum of all outcomes must be Exhaustive and Mutually exclusive.

8. Three approaches to probability: Classical: Probability is as simple as 1/n

9. Three approaches: Classical: Probability is as simple as 1/n Relative Frequency: long –run relative frequency

11. Three approaches: Classical: Probability is as simple as 1/n Relative Frequency: long –run relative frequency Subjective: the outcome as we perceive it, or believe it to be

12. Permutations and Combinations How many ways can n objects be arranged? n! = n(n-1)(n-2)… etc 4! = 4 X 3 X 2 X 1 = 24

13. There are 24 unique ways these people can be arranged as they wait.

14. Permutations and Combinations Permutation: How many ways can n objects be arranged r at a time?

15. How many ways can 4 people be arranged in unique order in groups of 2? = 24/2 = 12

16. Permutations and Combinations Combination: How many unique groups can be taken from n objects?

17. How many ways can 4 people be arranged in unique order in groups of 2? = 24/(2 X2) = 6

18. You and your friend Pat, along with two other people, show up for a job interview.

19. What is the probability that you will be selected first for the interview?

20. What is the probability that you will be selected first for the interview? What is the probability that the four of you will line up at random, and you are at the head of the line? Each of you are given a number (1 – 4), what is the probability that the four of you will line up at random, and the number given to each will be in the order of 1, 2, 3, 4?

21. What is the probability that you and Pat will be the first two interviewed?

22. What is the probability that you and Pat will be the first two interviewed? What is the probability that you and Pat will be interviewed first, and you will be interviewed before Pat?

23. Blaise Pascal 1623 - 1662 Pascal’s Triangle Binominal Coefficients

25. The coefficients are simply the combinations of n objects taken r at a time…

26. Suppose you wanted to know the probability of getting exactly 3 heads in ten tosses of a fair coin? The probability of a head is p = ½, the probability of a tail is q = ½.

27. Suppose you write a good resume, so good in fact that 80% of the recruiters who read it would offer you a job. Suppose further that you apply to a company that utilizes three independent resume assessors.

28. Suppose you write a good resume, so good in fact that 80% of the recruiters who read it would offer you a job. Suppose further that you apply to a company that utilizes three independent resume assessors. What is the probability of getting this job if all three assessors must recommend hiring?

29. Suppose you write a good resume, so good in fact that 80% of the recruiters who read it would offer you a job. Suppose further that you apply to a company that utilizes three independent resume assessors. What is the probability of getting this job if at least two out of three assessors must recommend hiring?

30. Some Interesting Problems A western city is 30% Hispanic, 45% of all convictions for spousal abuse are Hispanic. An activists claims that this is evidence that jurists are biased against Hispanics. Is this true?

31. Some Interesting Problems Gender Discrimination 1000 men and 1000 women applied to a university, 70% of the men were accepted, but only 40% of the women. Did gender discrimination take place? Yes! In fact…. The university discriminated against men!!

32. Some Interesting Problems Gender Discrimination 1000 men and 1000 women applied to a university, 70% of the men were accepted, but only 40% of the women. There were two programs: One excellent: 200 men 800 women applied One mediocre: 800 men 200 women applied

33. Some Interesting Problems Gender Discrimination 1000 men and 1000 women applied to a university, 70% of the men were accepted, but only 40% of the women. There were two programs: One excellent: 200 men 800 women applied One mediocre: 800 men 200 women applied Excellent program: 20% men 25% women accepted

34. Some Interesting Problems Gender Discrimination 1000 men and 1000 women applied to a university, 70% of the men were accepted, but only 40% of the women. There were two programs: One excellent: 200 men 800 women applied One mediocre: 800 men 200 women applied Mediocre program: 82.5% men 100% women accepted

35. Some Interesting Problems Gender Discrimination 1000 men and 100 women applied to a university, 70% of the men were accepted, but only 40% of the women. There were too programs: One excellent: 200 men 800 women applied One mediocre: 800 men 200 women applied In total, 700 men were accepted and only 200 women

36. Simpson’s Paradox http://en.wikipedia.org/wiki/Simpson%27s_paradox

37. Leniency vs Reciprocity Are employees who work for companies with higher pay more likely to be happy with their jobs? A study of 500 companies found no relationship between average employee pay and average workers’ happiness scores. The report concludes that higher paid workers are NOT more happy with their jobs. Is this true?

38. Some Interesting Problems A witness saw a hit-and-run. He claims the car was a city cab painted yellow. There was only one yellow cab in service at the time. Under the same conditions the witness was found to be able to see the difference between yellow and white cabs 80% of the time. The prosecutor tells the jury that they must convict the cab driver because the evidence is overwhelming. Is it?

39. It turns out that there is 20 yellow cabs in the city and 80 white cabs. What is the probability that the witness correctly identified the color of the car?

40. O. J. Simpson Trial

41. Johnnie Cochran argued that evidence that O.J. beat his wife was irrelevant because only 1 in 1,000 wife-beaters went on to kill their wives. What is the problem?

42. What is the rate of non-wife-beaters who kill their wives?

43. What is the rate of non-wife-beaters who kill their wives? If the rate is much smaller, then wife-beating is evidence.

44. What is the rate of non-wife-beaters who kill their wives? If the rate is much smaller, then wife-beating is evidence. 2. O.J.’s wife was actually murdered.

45. What is the rate of non-wife-beaters who kill their wives? If the rate is much smaller, then wife-beating is evidence. 2. O.J.’s wife was actually murdered. The real question is: If a wife is murdered, what is the probability that she had previously been beaten? From : Dr. Michael Starbird University of Texas

46. DNA Evidence. Suppose a murder was committed in southern California. DNA is found at the scene. A computer match finds a suspect.

47. DNA Evidence. Suppose a murder was committed in southern California. DNA is found at the scene. A computer match finds a suspect. The prosecuting attorney tells the jury that there is only one chance in a million that two people would have the same DNA evidence.

48. DNA Evidence. Suppose a murder was committed in southern California. DNA is found at the scene. A computer match finds a suspect. Therefore they must convict….

49. DNA Evidence. Suppose a murder was committed in southern California. DNA is found at the scene. A computer match finds a suspect. What is the actual probability that they have the real murderer based on this evidence?

50. An airport screening devise is so accurate that it will make an error only one time in 500,000! What will be the probability that a person accused of having a weapon is falsely accused?