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Bayes Theorem Motivating Example: Drug Tests A drug test gives a false positive 2% of the time (that is, 2% of those who test positive actually are not drug users) And the same test gives a false negative 1% of the time (that is, 1% of those who test negative actually are drug users) If Joe tests positive, what are the odds Joe is a drug user? Insufficient information! Suppose we know that 1% of the population uses the drug? Setting Up the Drug Test Problem • Let T be the set of people who test positive • Let D be the set of drug users • These are events, and Pr(T|D) is the probability that a drug user tests positive • Pr(T|D) = .99 because the false negative rate is 1%, that is, 99% of drug users test positive, 1% test negative • We want to know: What is Pr(D|T)? • This is a very different question: What is the probability you are a drug user, given that you test positive? Bayes Theorem Theorem: If Pr(A) and Pr(B) are both nonzero, Pr(A | B) Pr(A) Pr(B | A) Pr(B) Proof. We know that (*)Pr(A | B) Pr(B) Pr(A B) Pr(B | A) Pr(A) by the definition of conditional probability: Pr(A and B) similarly for Pr(B|A). Pr(A | B) Pr(B) , Then divide the left and right sides of (*) by Pr(B|A)∙Pr(B). Bayes, v. 2 Pr(A) Pr(B | A) Pr(A | B) Pr(B | A) Pr(A) Pr(B | A) Pr(A) This enables us to calculate Pr(A|B) using only the absolute probability Pr(A) and the conditional probabilities Pr(B|A) and Pr(B|¬A). Proof. We know that Pr(A | B) Pr(A) Pr(B | A) Pr(B) Now multiply by Pr(B|A) and rewrite Pr(B) using the law of total probability. Drug Test again • Suppose that a drug test has – 2% false positives (that is, 2% of the people who test positive are not drug users ) – 1% false negatives (1% of those who test negative are drug users). • Suppose 1% of the population uses drugs. If you test positive, what are the odds you are actually a drug user? Drug test, cont’d • Let D = “Uses drugs” • Let T = “Tests positive” Pr(D) .01 Pr(T | D) .02 Pr(T | D) .99 • What is Pr(D|T)? Pr(D) .01 Pr(T | D) .02 Pr(T | D) .99 Pr(D) Pr(T | D) Pr(D | T ) Pr(T | D) Pr(D) Pr(T | D) Pr(D) .01.99 .33 .99 .01 .02 .99 • If you fail the drug test, there is only one chance in three you are actually a drug user! • How can this be? Think about it. – Out of 1000 people there are 10 drug users and 990 non-users. – Of those 990, 2% or almost 20 test positive. – Almost all of the 10 users also test positive. – So there are 2 non-users for every user, among those who test positive! FINIS