People tend to intuitively overestimate the probability of being sick after having received a positive test result for a rare disease. This probability is calculated using Bayes's theorem. Understanding these probabilities can become easier using a contingency table.
Given properties of the test, the probability of the disease, and the size of the reference group, the table shows the number of people falling into the four different possible categories.
The conditional probability of being sick after having received a positive test result is simply the ratio of the people that are sick and that have tested positive to the total number of people having tested positive.