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n12 - CS 70 Spring 2012 Discrete Mathematics and...

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CS 70 Discrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 12 Conditional Probability A pharmaceutical company is marketing a new test for a certain medical disorder. According to clinical trials, the test has the following properties: 1. When applied to an affected person, the test comes up positive in 90% of cases, and negative in 10% (these are called “false negatives”). 2. When applied to a healthy person, the test comes up negative in 80% of cases, and positive in 20% (these are called “false positives”). Suppose that the incidence of the disorder in the US population is 5%. When a random person is tested and the test comes up positive, what is the probability that the person actually has the disorder? (Note that this is presumably not the same as the simple probability that a random person has the disorder, which is just 1 20 .) The implicit probability space here is the entire US population with uniform probabilities. This is an example of a conditional probability : we are interested in the probability that a person has the disorder (event A ) given that he/she tests positive (event B ). Let’s write this as Pr [ A | B ] . How should we define Pr [ A | B ] ? Well, since event B is guaranteed to happen, we should look not at the whole sample space Ω , but at the smaller sample space consisting only of the sample points in B . What should the conditional probabilities of these sample points be? If they all simply inherit their probabilities from Ω , then the sum of these probabilities will be ω B Pr [ ω ] = Pr [ B ] , which in general is less than 1. So we should normalize the probability of each sample point by 1 Pr [ B ] . I.e., for each sample point ω B , the new probability becomes Pr [ ω | B ] = Pr [ ω ] Pr [ B ] . Now it is clear how to define Pr [ A | B ] : namely, we just sum up these normalized probabilities over all sample points that lie in both A and B : Pr [ A | B ] : = ω A B Pr [ ω | B ] = ω A B Pr [ ω ] Pr [ B ] = Pr [ A B ] Pr [ B ] . Definition 12.1 (conditional probability) : For events A , B in the same probability space, such that Pr [ B ] > 0, the conditional probability of A given B is Pr [ A | B ] : = Pr [ A B ] Pr [ B ] . Let’s go back to our medical testing example. The sample space here consists of all people in the US — denote their number by N (so N 250 million). The population consists of four disjoint subsets: CS 70, Spring 2012, Note 12 1
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TP : the true positives (90% of N 20 = 9 N 200 of them); FP : the false positives (20% of 19 N 20 = 19 N 100 of them); TN : the true negatives (80% of 19 N 20 = 76 N 100 of them); FN : the false negatives (10% of N 20 = N 200 of them). Now let A be the event that a person chosen at random is affected, and B the event that he/she tests positive. Note that B is the union of the disjoint sets TP and FP , so | B | = | TP | + | FP | = 9 N 200 + 19 N 100 = 47 N 200 . Thus we have Pr [ A ] = 1 20 and Pr [ B ] = 47 200 .
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