Suppose we observe a random variableand wish to make inferences about another random variablewhereis drawn from some distribution

From the definition of conditional probability,

Again from the definition of conditional probability, we can express the joint probability by conditioning onto give

Substituting (2) into (1) gives Bayes’ theorem:

If there aremutually exclusive possible outcomes forthen we can write

hence

Bayes theorem gives rise to some surprises. Many people diagnosed with disease are falsely diagnosed. Suppose that one in a thousand adults has a disease. When an individual has a disease, a positive result will be returned 99% of the time, while a positive result will be returned for 2 % of individuals who do not have the disease. Let andthenandsoand

Less that one in twenty positive diagnoses are actually true positives. More than 95% of positives results are false positives.