Thinking about Probabilities

1. Thinking about Probabilities CCC8001

2. Assignment Watch episode 1 of season 1 of “Ancient Aliens.”

3. Assignment Find one thing that is said, shown, or presented in the episode that is misleading. I want you to describe it to me, then to explain why it is misleading.

4. Assignment What you describe to me can be misleading for any reason, not just a reason we’ve talked about in class. Just describe it, and tell me the reason why it is misleading.

5. Probability biases

6. Conjunction Fallacy Which of the following is most likely to happen? a. There will not be a final exam in this class. b. There will not be a final exam in this class, because the instructor has to leave the country. c. Lingnan University closes and there will not be a final exam in this class.

7. The Killer Ah Jong is a martial arts expert. He’s a high ranking member of the Triads, and he’s killed hundreds of people. Friends describe him as “dangerous.”

8. The Killer

9. What Is Most Likely? (a) Ah Jong sews women’s dresses. (b) Ah Jong sews women’s dresses so the police won’t think he’s a gangster (c) Ah Jong sews women’s dresses as part of his court-ordered rehabilitation.

10. Conjunction Fallacy Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable? a. Linda is a bank teller. b. Linda is a bank teller and is active in the feminist movement.

11. Conjunction Fallacy The correct answer is (a): a. There will not be a final exam. a. Ah Jong sews women’s dresses. a. Linda is a bank teller.

13. Conjunction Fallacy Suppose: 20,000 people meet the description (“31 years old, single, outspoken, and very bright. She majored in philosophy…”). How many of them are feminist bank tellers? Pick any number: 5,000.

14. Conjunction Fallacy How many of them are bank tellers (feminist or not feminist)? It has to be more than 5,000. That’s the number of feminist bank tellers. You have to add in the number of non-feminist bank tellers who meet the description. Let’s say that’s only 1 person.

15. Conjunction Fallacy Then the probability that Linda is a bank teller AND is active in the feminist movement is 5,000 out of 20,000 or 25%. And the probability that she is a bank teller [maybe a feminist, maybe not] is 5,001 out of 20,000 > 25%.

16. Mathematics Always, the probability of two events happening (Linda being a bank teller AND Linda being a feminist) is less than the probability of just one of those events happening (for example, Linda being a bank teller). The illusion that the opposite is true especially occurs in cases where one event explains the other.

17. George For example, suppose I tell you that there is a man named “George.” George turned water into wine, healed the sick, brought a dead person back to life, and came back to life himself after he died. What is the probability that George did all these things? How likely is it?

18. George You can say whatever you like. 1%, 10%, 99%. But suppose I add to the story. I say “George was the son of God. That’s why he had all these powers.”

19. George Many people will say that it’s more likely that George was the son of God AND did all these things than it is that he did all these things. But that can’t be true. “A & B” is always less (or equally) probable than A, or than B. For A & B to happen, A has to happen and also B has to happen.

20. Debiasing We can avoid this bias if we ask the question differently: There are 100 persons who fit the description above (that is, Linda’s). How many of them are: Bank tellers? ____ of 100 Bank tellers and active in the feminist movement? ____ of 100

21. Frequencies This shows that it’s good to translate percentages and probabilities into frequencies (number of X out of number of Y). We are less susceptible to representativeness bias when things are phrased in this way.

22. representativeness

23. Representativeness Our (false) judgment that Linda is more likely to be a feminist bank teller than to just be a bank teller is an example of how we judge the truth of claims based on how “representative” they are.

24. Representativeness Consider again our case of coin flips that seem non-random, due to clustering. Since coins land 50% heads and 50% tails, “XO” and “OX” are representative of this even split, whereas “XX” and “OO” don’t represent it. So sequences with clustering seem non-random, even if they are (random).

25. Representativeness Representativeness influences our other judgments as well. It’s hard to accept that two very tall parents tend to, on average, have less tall children (as regression to the mean requires). Children who are as tall as their parents are more representative of their parents’ heights.

26. Heuristics and Biases Representativeness is often a good heuristic. A heuristic is a strategy that is easy to use in problem solving but doesn’t always work when applied. There is often no good reason to distinguish between heuristics and biases.

27. Representativeness is a good heuristic (sometimes) because (sometimes) things are representative.

28. The Base rate neglect fallacy

29. Base Rate Fallacy • There are ½ million people in Russia are affected by HIV/ AIDS. • There are 150 million people in Russia.

30. Base Rate Fallacy Imagine that the government decides this is bad and that they should test everyone for HIV/ AIDS.

31. They develop a test with the following features: • If someone has HIV/ AIDS, then 95% of the time the test will be positive (correct), and only 5% of the time will it be negative (incorrect). • If someone does not have HIV/ AIDS, then 95% of the time the test will be negative (correct), and only 5% of the time will it be positive (incorrect).

32. The Test If someone has HIV/ AIDS, then : • 95% of the time the test will be positive (correct) • 5% of the time will it be negative (incorrect)

33. The Test If someone does not have HIV/ AIDS, then: • 95% of the time the test will be negative (correct) • 5% of the time will it be positive (incorrect)

34. You Get Tested Suppose you are a Russian who gets tested for HIV/ AIDS under the government program. The test comes out positive. How likely are you to have HIV/ AIDS?

36. True Positives We know that there are 500,000 people in Russia who have HIV/AIDS. How many will get a positive test result?

37. The Test If someone has HIV/ AIDS, then : • 95% of the time the test will be positive (correct) • 5% of the time will it be negative (incorrect)

38. True Positives 500,000 x 95% = 475,000

41. False Negatives? How many people who have HIV/AIDS will test negative? 500,000 – 475,000 = 25,000 500,000 x 5% = 25,000

44. True Negatives We also know that there are 150 million – 500,000 people in Russia who do not have HIV/AIDS. How many of them will correctly test negative?

45. The Test If someone does not have HIV/ AIDS, then: • 95% of the time the test will be negative (correct) • 5% of the time will it be positive (incorrect)

46. True Negatives (150,000,000 – 500,000) x 95% = ?

47. True Negatives 149,500,000 x 95% = ?

48. True Negatives 149,500,000 x 95% = 142,025,000