Bayes' Rule
This applet displays joint and conditional probabilities in a tree structure and computes the inverse probability according to Bayes' Rule. An important aspect of Bayes Rule is the effect of base rates on the usefulness of diagnostic tests. The initial values correspond to Exercise 125. The event B represents having the disease while its complement ~B represents the complment of B, which in this case is being healthy. Only 1% of the population is expected to have the disease B. A represents a positive test result, while A represents a negative test result. The probability of a postive test result if someone truly has the disease is 0.9 and the probability of a false positive test result if someone is healthy is 0.1. How useful is a positive test result? The total probability of a positive test result is represented by the two nodes ending with A; their total probability is 0.009 + 0.099 = 0.108. Note that most of the postiive test results come from the group ~B that does not have the disease. In fact, only 8.33% of those with positive test results are from the group B of people with the disease. Hence, the test is not very diagnostic.
Use the text boxes to change of any the probabilities. For example, how diagnostic would the test be if it were used in a hospital population in which we expected that half of the people would have the disease and half would not?