Maths prize puzzle for 22/12/17: boy or girl?

In a city where everyone wants a boy, each family continues having babies until they have a boy, then stops. Long term, what is the proportion of boys to girls in the country? Assume that at each birth the probability of boy or girl is the same.

Howard Tran and Mohaned al-Bassam won prizes.

The answer is – 50:50. The business about everyone wanting a boy makes no difference to the proportion. Why? Because each birth is an independent event, with P(girl) = 0.5 and P(boy) = 0.5, irrespective of what children the mother might have had before, and what she may want.

The conclusion is of some practical importance. In India, for example, the sex ratio of babies born into the best-off 20% of families is only 854 girls to 1000 boys. World-wide and generally, there is a slight excess of boy babies over girl babies, maybe an evolutionary response to lower life expectancy for males, but the general ratio is much nearer to equality than that 854:1000 figure.

The analysis shows that a figure like 854:1000 can develop only if there is selective abortion of female foetuses, or outright infanticide of unwanted girl babies.