In a country, all families want a boy. They keep having babies till a boy is born. What is the expected ratio of boys and girls in the country?

**Solution:**

Assumptions: Probability of having a boy or girl is same. Also, the probability of next kid being a boy doesn’t depend on history.

The problem can be solved by counting expected number of girls before a baby boy is born.

LetNGbe 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. NG can be written as sum of following infinite series. NG = 0*(1-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. NG = 0*(1/2) + 1*(1/2)^{2}+ 2*(1/2)^{3}+ 3*(1/2)^{4}+ 4*(1/2)^{5}+ ... 1/2*NG = 0*(1/2)^{2}+ 1*(1/2)^{3}+ 2*(1/2)^{4}+ 3*(1/2)^{5}+ 4*(1/2)^{6}+ ... NG - NG/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 NG/2 = (1/4)/(1-1/2) = 1/2 NG = 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

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