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From the Fun Fact files, here is a Fun Fact at the Advanced level:

Medical Tests and Bayes' Theorem

Suppose that you are worried that you might have a rare disease. You decide to get tested, and suppose that the testing methods for this disease are accurate 99 percent of the time (regardless of whether the results come back positive or negative). Suppose this disease is actually quite rare, occurring randomly in the general population in only one of every 10,000 people.

If your test results come back positive, what are your chances that you actually have the disease?

Do you think it is approximately: (a) .99, (b) .90, (c) .10, or (d) .01?

Surprisingly, the answer is (d), less than 1 percent chance that you have the disease!

Presentation Suggestions:
After discussing the reasons why the test results are not so reliable (below), see how changing the parameters affects the outcome. Would the result be so surprising if the disease were more common? How would things change if you allow the percentage of false positives and false negatives to be different?

The Math Behind the Fact:
This fact may be deduced using something called Bayes' theorem, which helps us find the probability of event A given event B, written P(A|B), in terms of the probability of B given A, written P(B|A), and the probabilities of A and B:

P(A|B)=P(A)P(B|A) / P(B)

In this case, event A is the event you have this disease, and event B is the event that you test positive. Here, P(B|A)=.99, P(A)=.0001, and P(B) may be derived by conditioning on whether event A does or does not occur:
P(B)=P(B|A)P(A)+P(B|not A)P(not A)
or .99*.0001+.01*.9999, which is less than 1 percent.

The basic reason we get such a surprising result is because the disease is so rare that the number of false positives greatly outnumbers the people who truly have the disease. This can be seen by thinking about what we can expect in 1 million cases. In those million, about 100 will have the disease, and about 99 of those cases will be correctly diagnosed as having it. Otherwise about 999,900 of the million will not have the disease, but of those cases about 9999 of those will be false positives (test results that are positive because of errors). So, if you test positive, then the likelihood that you actually have the disease is about 99/(99+9999), which gives the same fraction as above, approximately .0098 or less than 1 percent!

Note that these calculations wouldn't hold if the disease were not independently and identically distributed throughout the population (e.g., in the case of cancer due to familial tendency, environmental factor, asbestos exposure, etc.).

How to Cite this Page:
Su, Francis E., et al. "Medical Tests and Bayes' Theorem." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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Keywords:    probability
Subjects:    probability
Level:    Advanced
Fun Fact suggested by:   Francis Su
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