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 aare equal number of males and females.

.: Let ${E_1} =$ Selected person is a male

${E_2} =$ Selected person is a female

and $A =$ Selected person is grey haired.

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

$P\left( {A|{E_1}} \right) = \cfrac{5}{{100}} = \cfrac{1}{{20}}$ and $P\left( {A|{E_2}} \right) = \cfrac{{0.25}}{{100}} = \cfrac{1}{{400}}$

$\therefore$ Required probability $= P\left( {{E_1}|A} \right)$

$= \cfrac{{P\left( {{E_1}} \right) \cdot P\left( {A|{E_1}} \right)}}{{P\left( {{E_1}} \right)P\left( {A|,{E_1}} \right) + P\left( {{E_2}} \right)P\left( {A|{E_2}} \right)}}$

[Bayes' Theorem]
$= \cfrac{{\left( {\cfrac{1}{2}} \right)\left( {\cfrac{1}{{20}}} \right)}}{{\left( {\cfrac{1}{2}} \right)\left( {\cfrac{1}{{20}}} \right) + \left( {\cfrac{1}{2}} \right)\left( {\cfrac{1}{{400}}} \right)}}$

$= \cfrac{{\cfrac{1}{{20}}}}{{\cfrac{1}{{20}} + \cfrac{1}{{400}}}} = \cfrac{{20}}{{21}}$