Thinking and Deciding

152 DESCRIPTIVE THEORY OF PROBABILITY JUDGMENT
of view, we need to ask whether subjects can learn to take other information into
account, that is, whether they can learn more specific heuristics. The rate of errors
in the conjunction fallacy, at least, can be substantially reduced by brief instruction
in the logic of sets (Agnoli and Krantz, 1989). Such instruction is also effective in
the simpler version used with children (Agnoli, 1991). It thus appears that people
are capable of learning other heuristics, in at least some cases in which the representativeness
heuristic often leads to error.
In situations in which very careful thinking is required, such as medical decisions
potentially involving life and death, the evidence we have seen suggests that formal
calculations may be worthwhile. Even the best thinking — unaided by mathematics
— may give inferior results.
The gambler’s fallacy and probability matching
Another kind of error possibly related to the use of the representativeness heuristic
is the gambler’s fallacy, otherwise known as the law of averages. If you are playing
roulette and the last four spins of the wheel have led to the ball’s landing on black,
you may think that the next ball is more likely than otherwise to land on red. This
cannot be. The roulette wheel has no memory. The chance of black is just what it
always is. The reason people tend to think otherwise may be that they expect the
sequence of events to be representative of random sequences, and the typical random
sequence at roulette does not have five blacks in a row.
The gambler’s fallacy, or something like it, may affect people’s behavior when
they must make choices concerning repeated events. In a typical experiment on this
question, Peterson and Ulehla (1965) asked subjects to roll a die with four black
faces and two white faces and to predict (for monetary reward) which face would
be on top after each roll. Subjects should know that the die had no memory, so that
it was more likely to come up black each time, no matter how many times black
had come up in a row. But many subjects persisted in predicting white some of the
time, especially after a long run of blacks. This phenomenon was called “probability
matching” because people (and some animals, in similar tasks) approximately
matched the proportion of their responses to the probabilities of success. For example,
they would choose black 2/3 of the time. But this is only a rough approximation,
and individual subjects differ substantially. 4
Such behavior does not seem to result from boredom. Gal and Baron (1996)
asked subjects about hypothetical cases like this and found that many subjects believed
in the heuristic they were using, in much the way in which subjects believe in
naive theories of physics. In one case, for example, a die was rolled and the task was
to bet which color would be on top. A subject said, “Being the non-statistician I’d
keep guessing red as there are four faces red and only two green. Then after a number
of red came up in a row I’d figure, ‘it’s probably time for a green,’ and would predict
green.” Another subject seemed aware of the independence of successive trials but
4 Another explanation of these results, aside from representativeness, is that people are using a diversification
heuristic, of the same sort found in studies of choice, as described on p. 494.