Famous puzzle: Girl/Boy proportion problem (Sum of infinite series)

This is a trick question!

This question is very simple if you just learn to accept that most of the information given is completely irrelevant...

It doesnt matter how many families continue to have children and how many stop at 1 or 2...its no more relevant than what car they drive...

None of the information provided alters the statistical probability of a child born being male or female...its still 50%

Mike Scott is correct that you don't need to sum the series, but suppose you want to. Each family has 1 boy-that is easy. Each family has 50% chance of no girls, 25% chance of 1, etc. So the average number of girls is $$\sum_{i=0}^\infty \frac{i}{2^{i+1}}$$ The way to sum this is to remember that $$\sum_{i=0}^\infty a^{-i} = \frac{1}{1-1/a}=\frac a{a-1}$$ Now if you take the derivative with respect to a and evaluate it at a=2 So $$\frac{d}{da}\sum_{i=0}^\infty a^{-i} \\=\sum_{i=0}^\infty{-i}a^{-(i+1)} \\=\frac{d}{da} \frac{1}{1-1/a} \\=\frac d{da}\frac a{a-1} \\=\frac {a-1-a}{(a-1)^2} \\=\frac{-1}{(a-1)^2}=-1$$ for $a=2$. So there is an average of one girl per family as well

Google and other interviewing companies made an important assumption — that there are infinitely many families. Under this assumption, we have their clever and simple answer: In expectation, there are as many boys as girls.

However, it turns out that the answer depends on the number of families there are in the country. If there is a finite number of families, then in expectation, there'll be more boys than girls.

Consider the extreme scenario where there's only one family, then 1/2 the time the fraction of girls is 0 (B in our only family), 1/4 the time it's 1/2 (GB), 1/8 the time it's 2/3 (GGB), 1/16 the time it's 3/4 (GGGB) etc. And so the expected fraction of girls is:

$$\frac{1}{2}\times 0+\frac{1}{4}\times \frac{1}{2}+\frac{1}{8}\times \frac{2}{3}+\frac{1}{16}\times \frac{3}{4}+\dots = 1 - \ln 2 \approx 0.307.$$

If there are 4 families, the expected fraction of girls in the country is about 0.439; and if there are 10, it's about 0.475.

See this Math.Overflow answer for more details.

(By the way, this puzzle also appeared in Thomas Schelling's Micromotives and Macrobehavior, 1978, p. 72.)