The following experiment has been slightly modified for ease of blogging. You are given the following written description, which is assumed true:

Bill is 34 years old. He is intelligent, but unimaginative, compulsive, and generally lifeless. In school, he was strong in mathematics but weak in social studies and humanities.

No complaints about the description, please, this experiment was done in 1974. Anyway, we are interested in the probability of the following propositions, which may or may not be true, and are not mutually exclusive or exhaustive:

A: Bill is an accountant.
B: Bill is a physician who plays poker for a hobby.
C: Bill plays jazz for a hobby.
D: Bill is an architect.
E: Bill is an accountant who plays jazz for a hobby.
F: Bill climbs mountains for a hobby.

Take a moment before continuing to rank these six propositions by probability, starting with the most probable propositions and ending with the least probable propositions. Again, the starting description of Bill is assumed true, but the six propositions may be true or untrue (they are not additional evidence) and they are not assumed mutually exclusive or exhaustive.

In a very similar experiment conducted by Tversky and Kahneman (1982), 92% of 94 undergraduates at the University of British Columbia gave an ordering with A > E > C. That is, the vast majority of subjects indicated that Bill was more likely to be an accountant than an accountant who played jazz, and more likely to be an accountant who played jazz than a jazz player. The ranking E > C was also displayed by 83% of 32 grad students in the decision science program of Stanford Business School, all of whom had taken advanced courses in probability and statistics.

There is a certain logical problem with saying that Bill is more likely to be an account who plays jazz, than he is to play jazz. The conjunction rule of probability theory states that, for all X and Y, P(X&Y) <= P(Y). That is, the probability that X and Y are simultaneously true, is always less than or equal to the probability that Y is true. Violating this rule is called a conjunction fallacy.

Imagine a group of 100,000 people, all of whom fit Bill's description (except for the name, perhaps). If you take the subset of all these persons who play jazz, and the subs...

Bill_Smith

Bill_Smith

Bill_Smith

Bill_Smith