Base rate fallacy
The base rate fallacy, also called base rate neglect or base rate bias, is a formal fallacy. If presented with related base rate information (i.e. generic, general information) and specific information (information only pertaining to a certain case), the mind tends to ignore the former and focus on the latter.
Base rate neglect is a specific form of the more general extension neglect.
Many would answer as high as 0.95, but the correct probability is about 0.02.
To find the correct answer, one should use Bayes's theorem. The goal is to find the probability that the driver is drunk given that the breathalyzer indicated he/she is drunk, which can be represented as
where "D" means that the breathalyzer indicates that the driver is drunk. Bayes's theorem tells us that
We were told the following in the first paragraph:
As you can see from the formula, one needs p(D) for Bayes' theorem, which one can compute from the preceding values using
which gives
Plugging these numbers into Bayes' theorem, one finds that
A more intuitive explanation: on average, for every 1,000 drivers tested,
Therefore, the probability that one of the drivers among the 1 + 49.95 = 50.95 positive test results really is drunk is .
The validity of this result does, however, hinge on the validity of the initial assumption that the police officer stopped the driver truly at random, and not because of bad driving. If that or another non-arbitrary reason for stopping the driver was present, then the calculation also involves the probability of a drunk driver driving competently and a non-drunk driver driving (in-)competently.
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