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?