DECISION-MAKING THEORY AND RESEARCHprobability of one event occurring given the proba-
bility of another event occurring. These events are
sometimes called the cause and effect, or the
hypothesis and the data. Using H and D (hypothe-
sis and data) as the two events, Bayes’s Theorem is:
P(H|D) = P(H)P(D|H)/P(D) [1] or P(H|D) =
P(D|H)P(H)/[P(D|H)P(H)+P(D|−H)P(−H)] [2] That
is, the probability of H given D has occurred is
equal to [1] the probability of H occurring multi-
plied by the probability of D occurring given H has
occurred divided by the probability of D occur-
ring, or [2] the probability of D given H has
occurred multiplied by the probability of H then
divided by the probability of D given H has oc-
curred multiplied by the probability of H plus the
probability that D occurs given H has not occurred
multiplied by the probability that H does not occur.
The cab problem (introduced in Kahneman
and Tversky 1972) has been used in several studies
as a measure of whether people’s judgments are
consistent with Bayes’s Theorem. The problem is
as follows: A cab was involved in a hit-and-run
accident at night. Two cab companies, the Green
and the Blue, operate in the city. You are given the
following data: (a) 85 percent of the cabs in the city
are Green and 15 percent are Blue. (b) a witness
identified the cab as Blue. The court tested the
reliability of the witness under the same circum-
stances that existed on the night of the accident
and concluded that the witness correctly identified
each one of the two colors 80 percent of the time
and failed 20 percent of the time. What is the
probability that the cab involved in the accident
was Blue rather than Green? Using the provided
information and formula [2] above, P(Blue Cab|Wit-
ness says ‘‘Blue’’) = P(Witness says ‘‘Blue’’|Blue
Cab)P(Blue Cab)/[P(Witness says ‘‘Blue’’|Blue
Cab)P(Blue Cab) + P(Witness says ‘‘Blue’’|Green
Cab)P(Green Cab)] or P(Blue Cab|Witness says
‘‘Blue’’) = (.80)(.15)/[(.80)(.15)+(.20)(.85)] = (.12)/
[(.12)+(.17)] = .41
Thus, according to Bayes’s Theorem, the proba-
bility that the cab involved in the accident was
Blue, given the witness testifying it was Blue, is
0.41. So, despite the witness’s testimony that the
cab was Blue, it is more likely that the cab was
Green (0.59 probability), because the probabilities
for the base rates (85 percent of cabs are Green
and 15 percent Blue) are more extreme than those
for the witness’s accuracy (80 percent accuracy).
Generally, people will rate the likelihood that the
cab was Blue to be much higher than .41, and often
the response will be .80—the witness’s accuracy
rate (Tversky and Kahneman 1982).That finding has been used to argue that
people often ignore base rate information (the
proportions of each type of cab, in this case;
Tversky and Kahneman 1982), which is irrational.
However, other analyses of this situation are possi-
ble (cf. Birnbaum 1983; Gigerenzer and Hoffrage
1995), which suggest that people are not irrational-
ly ignoring base rate information. The issue of
rationality will be discussed further below.Expected Utility (EU) Theory. Bayes’s Theo-
rem is useful, but often we are faced with decisions
to choose one of several alternatives that have
uncertain outcomes. The best choice would be the
one that maximizes the outcome, as measured by
utility (i.e., how useful something is to a person).
Utility is not equal to money (high-priced goods
may be less useful than lower-priced goods), al-
though money may be used as a substitute meas-
ure of utility. EU Theory (von Neumann and
Morganstern 1947) states that people should maxi-
mize their EU when choosing among a set of
alternatives, as in: EU = ([Ui * Pi]; where Ui is the
utility for each alternative, i, and Pi is the probabili-
ty associated with that alternative.The earlier version of this theory (Expected
Value Theory, or EV) used money to measure the
worth of the alternatives, but utility recognizes
people may use more than money to evaluate the
alternatives. Regardless, in both EU and EV, the
probabilities are regarded similarly, so people on-
ly need consider total EU/EV, not the probability
involved in arriving at the total.However, research suggests that people con-
sider certain probabilities to be special, as their
judgments involving these probabilities are often
inconsistent with EU predictions. That is, people
seem to consider events that have probabilities of
1.0 or 0.0 differently than events that are uncertain
(probabilities other than 1.0 or 0.0). The special
consideration given to certain probabilities is called
the certainty effect (Kahneman and Tversky 1979).
To illustrate this effect, which of these two options
do you prefer? A. Winning $50 with probability .5
B. Winning $30 with probability .7 Now which of
these next two options do you prefer? C. Winning