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Question
A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability 0.10 of giving a (false) positive result when applied to a non-sufferer. It is estimated that 0.5% of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the probability that: given a positive result, the person is a sufferer
Solution
Let E1 ≡ the event that person is sufferer
E2 ≡ the event that person is not a sufferer
E1, E2 are mutually exclusive and exhaustive events
It is given that 0.5% of population are sufferers
∴ 99.5% of population are not sufferers
∴ P(E1) = `0.5/100` = 0.005
P(E2) = `99.5/100` = 0.995
Let T ≡ the event that result is positive
Since the test has a probability of 0.95 of giving positive results when person is sufferer and 0.10 when person is non-sufferer, we have
`"P"("T"//"E"_1)` = 0.95 and `"P"("T"//"E"_2)` = 0.10
By Baye's Theorem, the required probability
= `"P"("E"_1//"T")`
= `("P"("E"_1)*"P"("T"//"E"_1))/("P"("E"_1)*"P"("T"//"E"_1) + "P"("E"_2)*"P"("T"//"E"_2))`
= `(0.005 xx 0.95)/(0.005 xx 0.95 + 0.995 xx 0.1)`
= `(0.00475)/(0.10425)`.
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