Let the events M, F and G be defined as follows:
M: A male is selected
F: A female is selected
G: A person has grey hair
It is given that the number of males = the number of females
∴ P (M) = P (F) = $\frac{1}{2}$
Now, P (G/M) = Probability of selecting a grey haired person given that the person is a:
Male = 5% = $\frac{5}{100}$
Similarly, P (G/F) = 0.25% = $\frac{0.25}{100}$
A grey haired person is selected at random, the probability that this person is a male = P(M|G)
= [Using Baye’s Theorem]
= $\frac{\frac{1}{2}\times \frac{5}{100}}{\frac{1}{2}\times \frac{5}{100}+\frac{1}{2}\times \frac{0.25}{100}}$
= $\frac{\frac{5}{100}}{\frac{5}{100}+\frac{0.25}{100}}$
= $\frac{5}{5.25}$
= $\frac{20}{21}$