miscellaneous problem 3 on bayes theorem of probability

miscellaneous problem 3 on baye's theorem on probability for class xii of cbse , ncert, mathematics

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 are equal number of males and females.

Let E1 be the event that a male is selected.
Let E2 be the event that a female is selected.

Let A be the event that the selected person has grey hair,

Given that there are equal number of males and females

P(E1) = (1/2)
P(E2) = (1/2)

P(A/E1) = (5/100)  [given 5% of men have grey hair ]
P(A/E2) = (0.25/100)  [given 0.25% of women have grey hair ]

Required probability = P[ selected person is a male given that selected person has grey hair]

Required probability = p[E1/A]

                          P( E1 ) P( A / E1 )
P ( E1 / A ) = ---------------------------------------------
                P( E1 ) P( A / E1 ) +P( E2 ) P( A / E2 )


P ( E1 / A ) = [(1/2)(5/100)] / {[(1/2)(5/100)] + [(1/2)(0.25/100)]}

P ( E1 / A ) =  [5] / [5+0.25] = ( 100 / 105 ) = (20/21)

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