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Unformatted text preview: CS228 Problem Set #0 Solutions 1 CS 228, Winter 2008 Solutions to Problem Set #0: Probability Review 1. After your yearly checkup, the doctor has bad news and good news. The bad news is that you tested positive for a serious disease, and that the test is 99% accurate (i.e., the probability of testing positive given that you have the disease is 0.99, as is the probability of testing negative given that you don’t have the disease). The good news is that this is a rare disease, striking only one in 10,000 people. Why is it good news that the disease is rare? What are the chances that you actually have the disease? Answer: We are given the following information: P ( test 1 | disease 1 ) = . 99 P ( test | disease ) = . 99 P ( disease 1 ) = . 0001 test 1 where test 1 means that the test is positive. What the patient is concerned about is P ( disease 1 | test 1 ) . Roughly speaking, the reason it is a good thing that the disease is rare is that P ( disease 1 | test 1 ) is proportional to P ( disease 1 ) , so a lower prior for disease will mean a lower value for P ( disease | test ) . Roughly speaking, if 10,000 people take the test, we expect 1 to actu- ally have the disease, and most likely test positive, while the rest do not have the disease, but 1% of them (about 100 people) will test positive anyway, so P ( disease | test ) will be about 1 in 100. More precisely, using Bayes’ rule: P ( disease 1 | test 1 ) = P ( test 1 | disease 1 ) P ( disease 1 ) P ( test 1 | disease 1 ) P ( disease 1 ) + P ( test 1 | disease ) P ( disease ) = . 99 × . 0001 . 99 × . 0001 + 0 . 01 × . 9999 = . 009804 The moral is that when the disease is much rarer than the test accuracy, a positive test result...
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