# 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 - Mathematics and Statistics

Sum

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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