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Lecture #10- Bayes Theorem.doc

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Jeff Racine

Lecture # 10- Chapter 4 and 5 (Part 4) Bayes’ Theorem • Consider a random experiment having several possible mutually exclusive outcomes E 1..,E n Suppose that the probabilities of these outcomes, p(E )1 have been obtained We call these: • Prior Probabilities: they tell us the likelihood of basic outcomes prior to the information be given to us • Posterior Probability: They are conditional probabilities. Prior probability is now updated that you have the information o Ie: you buy a lottery digit and you hear on the radio and say you have the last 4 numbers. You can revise that probability once you are sure you know the numbers • By using Bayes’ Theorem, a prior probability is modified • The new probability is a posterior (conditional) probability Let A be the new information. By the definition of conditional probability, the posterior probability of the outcome E1 is given by Recall from the general multiplicative rule of probability that: Substituting (2) into (1) yields Bayes’ Theorem So we know that P(D)= 0.003, p(P|D)= 0.98, p(P|D), 0.99 and Bayes’ Theorem states: To determine Example (Bayes’ Theorem) • A researcher has devised a test to discover if newborn babies have a particular genetic disorder • The probability that a baby has this disorder is 0.003 (three tenths of one %) • If the baby does have the disorder, the test will be positive 98% of the time • If the baby does not have the disorder, the test will be 99% of the time • Assume the test is administered on a newborn baby and is positive • What is the probability that the baby has the disorder? o Let P= event that the test is positive and D= event that the baby has the disorder. Then by Bayes’ Theorem, p(D|P)= p(D) x p(P|D)/ p(P) (P|D)= Probability given that you test positive D = you don’t have the disorder
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