The conjunction fallacy is a logical fallacy that occurs when it is assumed that specific conditions are more probable than a single general one.
The most oftcited example of this fallacy originated with Amos Tversky and Daniel Kahneman:^{[1]}
85% of those asked chose option 2.^{[2]} However the probability of two events occurring together (in "conjunction") is always less than or equal to the probability of either one occurring alone—formally, for two events A and B this inequality could be written as , and
For example, even choosing a very low probability of Linda being a bank teller, say Pr(Linda is a bank teller) = .05 and a high probability that she would be a feminist, say Pr(Linda is a feminist) = .95, then, assuming independence, Pr(Linda is a bank teller AND Linda is a feminist) = .05 × .95 or .0475, lower than Pr(Linda is a bank teller).
Tversky and Kahneman argue that most people get this problem wrong because they use the representativeness heuristic to make this kind of judgment: Option 2 seems more "representative" of Linda based on the description of her, even though it is clearly mathematically less likely.
(As a side issue, some people may simply be confused by the difference between 'and' and 'or'. Such confusions are often seen in those who have not studied logic, and the probability of such sentences using 'or' instead of 'and' is completely different. They may infer sentence #1 assumes Linda is necessarily not active in the feminist movement.)
Many other demonstrations of this error have been studied. In another experiment, for instance, policy experts were asked to rate the probability that the Soviet Union would invade Poland, and the United States would break off diplomatic relations, all in the following year. They rated it on average as having a 4% probability of occurring. Another group of experts was asked to rate the probability simply that the United States would break off relations with the Soviet Union in the following year. They gave it an average probability of only 1%. Researchers argued that a detailed, specific scenario seemed more likely because of the representativeness heuristic, but each added detail would actually make the scenario less and less likely.^{[3]} In this way it could be similar to the misleading vividness or slippery slope fallacies, though it is possible that people underestimate the general possibility of an event occurring when not given a plausible scenario to ponder.
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In a recent work of Franco,^{[4]} the
conjunction fallacy has been described with the mathematical
formalism of quantum mechanics. In particular, it has been
shown that each couple of mutually exclusive events
(Linda is/isn't feminist, or Linda is/isn't bankteller) can be
associated to a basis in a 2dimensional vector space.
Moreover, it is assumed that the subject's beliefs about such
events is described by a vector (called opinion state  s > ), which can be written as
a superposition of the basis vectors:
 s > = s_{0} 
a_{0} > + s_{1} 
a_{1} >
where  a_{0} >
and  a_{1} > are
the basis vectors relevant to a particular couple of mutually
exclusive events (for example, Linda is/isn't feminist). The
subjective probability relevant to event a_{1} (Linda IS feminist)
is
P(a_{1}) = 
s_{1}  ^{2}
If we want to describe the subjective probability relevant to
another couple of mutually exclusive events b_{0},b_{1}
(Linda is/isn't a bankteller), the law of total probability is
replaced in the quantum framework by the following law:
P(b_{1}) =
P(a_{1})P(b_{1}
 a_{1}) +
P(a_{0})P(b_{1}
 a_{0}) +
Interfe
rence
where the interference term (with a precise mathematical form) has
a very important role in the conjunction fallacy. In fact, the
presence of strongly negative interference terms can make P(b_{1}) <
P(a_{1})P(b_{1}
 a_{1}), which is precisely the
conjunction fallacy (the estimated probability that Linda is
bankteller is lower than the estimated probability that Linda is
feminist and bankteller).
An important fact is that in quantum mechanics it is impossible to
measure simultaneously two noncommuting observable quantities.
Thus the joint probability is replaced by the concept of
consecutive probability P(a_{1})P(b_{
1}  a_{1}).
In other words, it has been shown that the use of quantum
probability allows to describe in a natural way the conjunction
fallacy.
