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In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same)

Answer : The problem can be solved by counting expected number of girls before a baby boy is born. Let G be the expected no. of girls before a boy is born. Let p be the probability that a child is girl and (1-p) be probability that a child is boy. G can be written as sum of following infinite series.

G = 0*p + 1*p*(1-p) + 2*p*p*(1-p) + 3*p*p*p*(1-p) + 4*p*p*p*p*(1-p) +..... Putting p = 1/2 and (1-p) = 1/2 in above formula. G = 0*(1/2) + 1*(1/2)2 + 2*(1/2)3 + 3*(1/2)4 + 4*(1/2)5 + ... 1/2*G = 0*(1/2)2 + 1*(1/2)3 + 2*(1/2)4 + 3*(1/2)5 + 4*(1/2)6 + ... G - G/2 = 1*(1/2)2 + 1*(1/2)3 + 1*(1/2)4 + 1*(1/2)5 + 1*(1/2)6 + ...

Using sum formula of infinite geometrical progression with ratio less than 1

=> G/2 = (1/4)/(1-1/2) = 1/2 => G = 1

So Expected Number of number of girls = 1. Since the expected number of girls is 1 and there is always a baby boy, the expected ratio of boys and girls is 50:50