Question

Suppose that 5% of men and 0.25% of women have grey hair. A grey-haired person is selected at random. What is the probability of this person being male?

Assume that there are equal number of males and females.

Answer 1

Suppose there are the same number of males.

Event E₁ = Birth of a man, E₂ = Birth of a woman

A = having grey hair

∴ P(E₁) = `1/2`, P(E₂) = `1/2`

`P(A/E_1)` = 5% = 0.05

`P(A/E_2)` = 0.25% = 0.0025

Hence, by Bayes' theorem,

`P((E_1)/A) = (P(E_1) xx P(A/E_1))/(P(E_1) xx P(A/E_1) + P(E_2) xx P(A/E_2)`

= `(1/2 xx 0.05)/(1/2 xx 0.05 + 1/2 xx 0.0025)`

= `500/(500 + 25)` 

= `500/525`

= `20/21`