Fig. 1 The sample space S with events A and B
On first pass, Bayes rule would seem to be just a trivial extension of the concept of conditional probability. I begin by reviewing the definition of conditional probability.
Let Fig. 1 represent a sample space S consisting of 12 equally likely sample points. The events A and B are the subsets of sample points shown in the figure. Under the equally likely assumption we have the following.
Conditional probabilities effectively change the sample space. For instance, P(A|B) restricts the sample space to B and is calculated under the equally likely scenario by counting up the number of sample points that are both in A and B and dividing by the number of sample points in B. Formally we have
An example of the use of Bayes' theorem is the evaluation of drug test results. Suppose a certain drug test is 99% sensitive and 99% specific, that is, the test will correctly identify a drug user as testing positive 99% of the time, and will correctly identify a non-user as testing negative 99% of the time. This would seem to be a relatively accurate test, but Bayes' theorem can be used to demonstrate the relatively high probability of misclassifying non-users as users. Let's assume a corporation decides to test its employees for drug use, and that only 0.5% of the employees actually use the drug. What is the probability that, given a positive drug test, an employee is actually a drug user?