[ Home ] [ Up ] [ Answer ] [ Other Examples ]

The answer to this problem uses Bayes' theorem.

Let P(H) be the numerical probability that the hypothesis H is correct in the absence of any evidence--the prior probability. In the above example, H is the hypothesis that you have the disease and P(H) is 0.01 (1%).

You then take the test and obtain a positive outcome; this is the evidence E.

Let P(H | E) be the probability that H is correct given the evidence E. This is the revised estimate you want to calculate.

Let P(E | H) be the probability that E would be found if indeed H occurred. In the example, the test always detects disease when it is present, so:

P(E | H) = 1

in this case -- that is, 100%.

To compute the new estimate, you first calculate P(H_wrong), the probability that H does not occur, which is 0.99 in our example. Then, you calculate P(E | H_wrong), the probability that the evidence E would be found (i.e., the test comes out positive) even though H did not occur (i.e., you do not have the disease), which is 0.21 in the example.

Bayes' theorem says that:

or 4.6% !

Intuition Fails

This result defies most people's expectation.  Humans are not very good at recognizing such situations.

This is but one example of many with common characteristics:

•  The test is relatively good

but

•  The incidence of the disease is very low

The latter is the key point.

Example source: http://www.maa.org/devlin/devlin_2_00.html