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# n16 - CS 70 Discrete Mathematics for CS Spring 2008 David...

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Unformatted text preview: CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 16 Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. 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 condition 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 condition? (Note that this is presumably not the same as the simple probability that a random person has the condition, which is just 1 20 .) This is an example of a conditional probability : we are interested in the probability that a person has the condition (event A ) given that he/she tests positive (event B ). Let’s write this as Pr [ A | B ] . How should we compute Pr [ A | B ] ? Well, since event B is guaranteed to happen, we need to 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 probabilities of these sample points be? If they all simply inherit their probabilities from Ω , then the sum of these probabilities will be ∑ ϖ ∈ B Pr [ ϖ ] = Pr [ B ] , which in general is less than 1. So we need to scale the probability of each sample point by 1 Pr [ B ] . In other words, for each sample point ϖ ∈ B , the new probability becomes Pr [ ϖ | B ] = Pr [ ϖ ] Pr [ B ] . Now it is clear how to compute Pr [ A | B ] : namely, we just sum up these scaled probabilities over all sample points that lie in both A and B : Pr [ A | B ] = ∑ ϖ ∈ A ∩ B Pr [ ϖ | B ] = ∑ ϖ ∈ A ∩ B Pr [ ϖ ] Pr [ B ] = Pr [ A ∩ B ] Pr [ B ] . Definition (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. Let N denote the number of people in the US (so N ≈ 250 million). The population consists of four disjoint subsets: CS 70, Spring 2008, Note 16 1 T P : the true positives (90% of N 20 , i.e., 9 N 200 of them); FP : the false positives (20% of 19 N 20 , i.e., 19 N 100 of them); T N : the true negatives (80% of 19 N 20 , i.e., 76 N 100 of them); FN : the false negatives (10% of N 20 , i.e., N 200 of them). We choose a person at random. Recall that A is the event that the person so chosen is affected, and B the event that he/she tests positive. Note that B is the union of the disjoint sets T P and FP , so | B | = | T P | + | FP | = 9 N 200 + 19 N 100 = 47 N 200 ....
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n16 - CS 70 Discrete Mathematics for CS Spring 2008 David...

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