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

Let $E_1$: the evnet that selected person is a male and $E_2$ : the event that selected person is a female.

$E_{1}and{E}_2$ are mutually exclusive and exhaustive events. Moreover,

$P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2}$

Let E: the event that selected person is grey haired.

$\therefore P\left(\frac{E}{E_1}\right)=\frac{5}{100}=\frac{1}{20}$

and P$\left(\frac{E}{E_2}\right)=\frac{0.25}{100}=\frac{1}{400}$

By using Baye's theorem

$Requiredprobability=P\left(\frac{E}{E_1}\right)=\frac{P\left(\frac{E}{E_1}\right)P\left(E_1\right)}{P\left(\frac{E}{E_1}\right)P\left(E_1\right)+P\left(\frac{E}{E_2}\right)P\left(E_2\right)}=\frac{\frac{1}{20}\times 12}{\frac{1}{20}\times \frac{1}{2}+\frac{1}{400}\times \frac{1}{2}}=\frac{\frac{1}{20}}{\frac{1}{20}+\frac{1}{400}}=\frac{\frac{1}{20}}{\frac{21}{400}}=\frac{20}{21}$