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Old 08-05-2007   #1
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Default probability (expected value) help pls...?

R={test result is positive}, D={person has the disease}.Assume that for a randomly selected person, P(D) = 0.02, P(R|D) = 1, P(R| not D) = 0.05, so that the inexpensive test only gives false positive, and not false negative, results.Suppose that this inexpensive test costs $10. If a person tests positive then they are also given a more expensive test, costing $100, which correctly identifies all persons with the disease . What is the expected cost per person if a population is tested for the disease using the inexpensive test followed, if necessary, by expensive test?
Old 08-05-2007   #2
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It would be helpful to find out the probability of administering the expensive test to any person. Call this P(E).P(E) = P(E|D)P(D) + P(E|~D)P(~D) as you can check.(~ means not).What's P(E|D)? This is P(R|D) since if they have the disease then the result of the first test is positive and so they are given the expensive test.What's P(E|~D)? This is P(R|~D) since if they don't have the disease, the probability of getting the expensive test is the probability that the first test result is positive.So we can calculate P(E):P(E) = P(E|D)P(D) + P(E|~D)P(~D)= 1*0.02 + 0.05*0.98= 0.0686.Using this we can compute the expected cost of testing a person.It costs at least $10 to test each person, and with probability P(E) we'll have to spend another $100. So the expected cost is 10 + P(E)*100 = 16.86.

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