Suppose 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.

Solution

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

$= P (M | G) = \frac{P (M) \times P (G | M)}{P (M) \times P (G | M) + P (F) \times P (G | F)} . . . . . . [Using Baye' s Theorem] = \frac{\frac{1}{2} \times \frac{5}{100}}{\frac{1}{2} \times \frac{5}{100} + \frac{1}{2} \times \frac{0.25}{100}} = \frac{\frac{5}{100}}{\frac{5}{100} + \frac{0.25}{100}} = \frac{5}{5.25} = \frac{20}{11}$

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Probability

Suppose 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.

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