Extended Bayes' Theorem
If are mutually exclusive and exhaustive events, then:
Example: Medical Testing
A disease affects 1% of a population. A test for the disease has 95% sensitivity (correctly identifies diseased) and 90% specificity (correctly identifies non-diseased). If a person tests positive, what is the probability they have the disease?
Example: Urn Problem
Urn I contains 3 red and 4 blue balls. Urn II contains 5 red and 6 blue balls. A ball is drawn at random from a randomly chosen urn. If the ball is red, find the probability it came from Urn I.
Example: Factory Problem
Factory A produces 60% of items, Factory B produces 40%. Defect rates are 2% and 3% respectively. If an item is defective, find the probability it came from Factory A.