Question

Suppose $5\%$ men and $0.25\%$ women has white hairs chosen at random. Find the probability that selected person is a male. Assume that these are equal number of men and women

Answer 1

Given that number of men and women are equal
Let event $E_1=$ to be a man
$E_2=$ to be woman
$A=$ white hair
Probability to choose one man $P\left(E_1\right)=\frac{1}{2}$
Probability to choose a woman $P\left(E_2\right)=\frac{1}{2}$
Probability that a man has white hair
$P\left(\frac{A}{E_1}\right)=5$% $=\frac{5}{100}=0.05$
Probability that a woman has white hair
$P\left(\frac{A}{E_2}\right)=0.25$% $=\frac{25}{100}=0.25$
Now probability that male is selected
$P\left(\frac{E_1}{A}\right)=\frac{P\left(E_1\right).P\left(\frac{A}{E_1}\right)}{P\left(E_1\right)\times P\left(\frac{A}{E_1}\right)+P\left(E_2\right)\times P\left(\frac{A}{E_2}\right)}$
$=\frac{\frac{1}{2}\times 0.05}{\frac{1}{2}\times 0.05+\frac{1}{2}\times 0.0025}=\frac{0.05}{0.05+0.0025}=\frac{0.05}{0.525}=\frac{500}{525}=\frac{20}{21}$
Hence required probability$=\frac{20}{21}$