Let be the event Person X has disease Z. Lt moreover assume that Z is a rare dia
ID: 3064500 • Letter: L
Question
Let be the event Person X has disease Z. Lt moreover assume that Z is a rare diaase, Le., the probability that a X has discase Z is 0.01. Let us assume a certain diagnostic exam is performed on X and it indicates that X indeed has discase Z. However, the exam is not perfect, .e., it has a false positive rate of 0.1, .e., there is 0.1 probability that the exam determines X has the discase, even though this is not the case. Moreover, it has a missed detection probability of 0.05, i.c., there is a 0.05 probability that the exam determines that X does not have the discase when instead X has indeed the discase. In the light of the exam result, what is the posterior probability that X has discase Z? If the exam is performed twice, and both times it indicates that person X has the discase, would the probability change? If your answer is "yes explain why and how it will change. If your answer is "no" explain why it will not change.Explanation / Answer
Baye's theorem - P(A | B) = P (A and B)/P(B)
P(test indicating X has disease) = P(test indicating X has disease when he actually has disease) + P(test indicating X has disease when he is not having disease)
P(X has disease Z when the test indicates that X has disease) = P(X has disease and test indicates the same)/P(test indicates X has disease)
= 0.01x1/(0.01x1 + 0.99x0.1)
= 0.0917
If the exam is performed twice, and both times its indicated that the person has disease, the probability that he actually has disease
= P(having disease and both times test indicating he has)/P(both times test indicating he has)
= 0.01/(0.01 + 0.99x0.1x0.1)
= 0.5025
so, the probability will change (explanation is the calculation of probability given above)
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