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 subset of all these persons who play jazz and are accountants, the second subset will always be smaller because it is strictly contained within the first subset.
Could the conjunction fallacy rest on students interpreting the experimental instructions in an unexpected way - misunderstanding, perhaps, what is meant by "probable"? Here's another experiment, Tversky and Kahneman (1983), played by 125 undergraduates at UBC and Stanford for real money:
Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequences of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you chose appears on successive rolls of the die. Please check the sequence of greens and reds on which you prefer to bet.
1. RGRRR
2. GRGRRR
3. GRRRRR
65% of the subjects chose sequence 2, which is most representative of the die, since the die is mostly green and sequence 2 contains the greatest proportion of green rolls. However, sequence 1 dominates sequence 2, because sequence 1 is strictly included in 2. 2 is 1 preceded by a G; that is, 2 is the conjunction of an initial G with 1. This clears up possible misunderstandings of "probability", since the goal was simply to get the $25.
Another experiment from Tversky and Kahneman (1983) was conducted at the Second International Congress on Forecasting in July of 1982. The experimental subjects were 115 professional analysts, employed by industry, universities, or research institutes. Two different experimental groups were respectively asked to rate the probability of two different statements, each group seeing only one statement:
- "A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983."
- "A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983."
Estimates of probability were low for both statements, but significantly lower for the first group than the second (p < .01 by Mann-Whitney). Since each experimental group only saw one statement, there is no possibility that the first group interpreted (1) to mean "suspension but no invasion".
The moral? Adding more detail or extra assumptions can make an event seem more plausible, even though the event necessarily becomes less probable.
Do you have a favorite futurist? How many details do they tack onto their amazing, futuristic predictions?
Tversky, A. and Kahneman, D. 1982. Judgments of and by representativeness. Pp 84-98 in Kahneman, D., Slovic, P., and Tversky, A., eds. Judgment under uncertainty: Heuristics and biases. New York: Cambridge University Press.
Tversky, A. and Kahneman, D. 1983. Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90: 293-315.
I think this might possibly be explained if they looked at it in reverse. Not "how likely is it that somebody with this description would be A-F", but "how likely is it that somebody who's A-F would fit this description".
When I answered it I started out by guessing how many doctors there were relative to accountants -- I thought fewer -- and how many architects there were relative to doctors -- much fewer. If there just aren't many architects out there than it would take a whole lot of selection for somebody to be more likely to be one.
But if you look at it the other way around then the number of architects is irrelevant. If you ask how likely is it an architect would fit that description, you don't care how many architects there are.
So it might seem unlikely that a jazz hobbyist would be unimaginative and lifeless. But more likely if he's also an accountant.
I think this is a key point - given a list of choices, people compare each one to the original statement and say "how well does this fit?" I certainly started that way before an instinct about multiple conditions kicked in. Given that, its not that people are incorrectly finding the chance that A-F are true given the description, but that they are correctly finding the chance that the description is true, given one of A-F.
I think the other circumstances might display tweaked version of the same forces, also. For example, answering the suspension of relations question not as P(X^Y) vs P(Y), but perceiving it as P(Y), given X.