Grinstead-Snell-Probability · Problem 4.1.18
Conditional Probability
Mathematics · bayes-formula-diseases
Problem
A doctor assumes that a patient has one of three diseases , , or . Before any test, he assumes an equal probability for each disease. He carries out a test that will be positive with probability if the patient has , if he has disease , and if he has disease . Given that the outcome of the test was positive, what probabilities should the doctor now assign to the three possible diseases?Intuitive explanation
This is a classic application of Bayes' theorem, where you update your initial beliefs (prior probabilities) using new evidence (the test result). The key insight is that a positive test is more likely to occur if the patient has than the other diseases, so you should shift your probability mass toward . You'll multiply each prior by its corresponding likelihood, then normalize so the three probabilities sum to 1.
Intuitive explanation
This is a classic application of Bayes' theorem, where you update your initial beliefs (prior probabilities) using new evidence (the test result). The key insight is that a positive test is more likely to occur if the patient has than the other diseases, so you should shift your probability mass toward . You'll multiply each prior by its corresponding likelihood, then normalize so the three probabilities sum to 1.