11/20/2014
For those of you that participated in the rationality activity this week, I did finally tally the conjunction fallacy results. Congratulations! You're rational!
The conjunction fallacy occurs when probabilities are estimated in a way such that P(A) < P(A∧B). That is, if you estimate the probability that A as being less than the probability that A and also B, you're committing the conjunction fallacy. This often arises when people employ stereotypes to estimate probabilities about people. For example, suppose that Stanley is quiet, keeps to himself, and is really into math. Now think about the following statements about Stanley:
1.) Stanley enjoys watching pro football
2.) Stanley is an accountant and enjoys watching pro football
Statement 1 is more likely than 2, because 1 is contained within 2, but many people will rank 2 as being more probable.
We had a "quiz" that was intended to gauge your ability to estimate probabilities rationally. Okay, really it was just intended to get you thinking about estimating probabilities before I talked about it, but the results are interesting too. It consisted of a series of statements, some of which were contained within the others, and you were supposed to say how much you'd be willing to bet on either $10, $100, or $10000. Here are the average ratios of your answers, for each A, A∧B pair, that is, the average P(A) / P(A∧B) (so > or = does not violate conjunction):
Obama disapproval / Obama achieves nothing and disapproval: 1.2
Rick and Robert argue / Rick and Robert argue with details: 1.33
Coin toss: HTHHH / THTHHH: 1.5
Smoking rates / smoking rates and cigarette tax: 1.2
Get hit by a meteor / get hit by a meteor once in each hand: 2500
All of these are greater than one (good), and nobody answered with a ratio less than one for any of the answers (really good!). I think the whole thing was put together awkwardly, and if I did it again, I'd make some changes. I do not think the questions were ideally phrased to get irrational answers, mainly because I did not employ the stereotypes about people, which tend to get strong results. I avoided this mainly because I wanted to use the betting method to get probabilities, and betting on whether Stanley is into pro football just seemed distractedly strange. Still, based on more careful studies, I expect that if you gave this to the general public you'd get a lot of people answering with ratios less than one.
Anyway, thanks to those of you that participated, and thanks for putting up with my presentation that was a little rough around the edges.