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 E_{1} ≡ the event that person is sufferer

E_{2} ≡ the event that person is not a sufferer

E_{1}, E_{2} are mutually exclusive and exhaustive events

It is given that 0.5% of population are sufferers

∴ 99.5% of population are not sufferers

∴ P(E_{1}) = `0.5/100` = 0.005

P(E_{2}) = `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)`.