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Home » Suppose of men and 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 is an equal number of males and females.
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 is an equal number of males and females.
Bayes’s Theorem and its Applications | CBSE | Class 12 | Exercise 30A | Maths | RS Aggarwal
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 is an equal number of males and females.

Let MG : Men having grey hair
WG: Women having grey hair
G: Having grey hair
Given an equal number of males and females. So let’s assume both the probability be \frac{1}{2}
We want to find P(M G \mid G), i.e. probability of a randomly selected grey person to be male
\begin{array}{l} P(M G \mid G)=\frac{P(M G) \cdot P(G \mid M G)}{P(M G) \cdot P(G \mid M G)+P(W G) \cdot P(G \mid W G)} \\ =\frac{\left(\frac{1}{2}\right)\left(\frac{5}{100}\right)}{\left(\frac{1}{2}\right)\left(\frac{5}{100}\right)+\left(\frac{1}{2}\right)\left(\frac{0.25}{100}\right)} \\ =\frac{5}{5.25} \\ =\frac{20}{21} \end{array}
Therefore, the probability of a randomly selected grey person to be male is \frac{20}{21}

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