Question

Suppose a couple will continue having children until they have at least two children of each sex (two boys and two girls). How many children might they expect to have? For @inkyvoyd @Rohangrr

Answer 1

Interesting... an expected value problem with infinite outcomes...

Answer 2

Let \(X\) be the number of children they have. Clearly: \[
\Pr(X<2) = 0
\]So we consider... \(2\leq X\leq \infty\)

Answer 3

We want to find \[
\Pr (X=n)
\]Consider if they have 4 children...

Answer 4

3 girls and a boy or 3 boys and a girl...

Answer 5

Now order does matter... so we have: \[
\Pr(X=4) = (0.5)^3(0.5)+(0.5)^3(0.5) = (0.5)^3
\]It seems like our general formula is going to be: \[
\Pr(X=n)=(0.5)^{n-1} = \frac{1}{2^{n-1}}
\]

Answer 6

this would leave our expected value at: \[ \Large
E[X] = \sum_{n=2}^{\infty} n\Pr(X=n) =\sum_{n=2}^{\infty} \frac{n}{2^{n-1}}
\]