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 negative result, the person is a non-sufferer

#### Solution

Let event T: Test positive

event S: Sufferer

P(S) = `0.5/100` = 0.005

∴ P(S') = 1 – P(S) = 1 – 0.005 = 0.995

Since a probability of getting a positive result when applied to a person suffering from a disease is 0.95 and the probability of getting a positive result when applied to a non-sufferer is 0.10.

∴ `"P"("T"/"S")` = 0.95 and `"P"("T"/"S'")` = 0.10

∴ P(T) = `"P"("S") "P"("T"/"S") + "P"("S'") "P"("T"/"S'")`

= 0.005 × 0.95 + 0.995 × 0.10

= 0.10425

∴ P(T') = 1 – P(T) = 1 – 0.10425 = 0.8958

`"P"("T'"/"S'")` = 1 – 0.1 = 0.9

Required probability = `"P"("S'"/"T'")`

By Bayes’ theorem,

`"P"("S'"/"T'") = ("P"("S'")"P"("T'"/"S'"))/("P"("T'"))`

= `(0.995 xx 0.9)/0.8958`

= `0.8955/0.8958`