A couple decides to keep having children until
- Cond1: they have at least one boy and at least one girl,
- Cond2: with boy older than girl
and then stop. Assume they never have twins, that the “trials” are independent with probability $1/2$ of a boy, and that they are fertile enough to keep producing children indefinitely. What is the expected number of children?
Note: updated
If we consider just Cond1, answer would be
Let $X$ be the number of children needed, starting with the 2nd child, to obtain one whose gender is not the same as that of the firstborn. Then $X − 1$ is Geom(1/2), so $E(X) = 2$. This does not include the firstborn, so the expected total number of children is $E(X + 1) = E(X) + 1 = 3$.
Best Answer
Hint: