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This is Bayes' theorem. The theorem is attributed to the Reverend [[Thomas Bayes]] (1702-1761) an English nonconformist minister, but it is likely to have been known to earlier mathematicians such as [[Pierre Laplace]].
This is Bayes' theorem. The theorem is attributed to the Reverend [[Thomas Bayes]] (1702-1761) an English nonconformist minister, but it is likely to have been known to earlier mathematicians such as [[Pierre Laplace]].
The theorem has many applications.
Example 1.
Suppose a particular disease afflicts 1% of the population. Suppose that a test for the disease is 95% accurate. Suppose that someone tests positive for the disease but there is no other evidence that they have the disease. What is the probability that they have the disease?
Let A be the event that the test result is positive. Let B be the event that the person actually has the disease.
Before the test result is known our probability for the person having the disease is p(B) = 1% = 0.01. The probability that the person has a positive test result, giving that they have the disease, is 95% = 0.95. The denominator term, p(A) is a little more complex since A can occur in two different ways. If the person has the disease, they will test positive with probability 0.95. If the person does not have the disease they will test positive with probability 5%. We denote the probability that B is not true by B'. The laws of probability require that B' = 1 - B. Thus
p(A) = p(B)p(A|B) + p(B')p(A|B') = 0.01 x 0.95 + 0.99 x 0.05 = 0.059
Hence
p(B|A) 0.01 x 0.95/0.059 = 0.16
In other words, there is only a 16% chance that a person testing positive actually has the disease.