A family decides to have children until it has three kids of the same gender. Assuming P(B) = P(G) = 0.5, what is the pmf of X = number of children in the family?

Question

anonymous · Answer

I set up the pmf to look like this: nb(x; 3, 0.5) 

I have no clue how to find x though.

reemii · Answer

The definition of $X$ is : "number of children in the family" given that the parents make children until there are 3 of the same gender.
You can't really identity the pmf from the beginning: it is not you that will fix it to "nb(3,0.5)". Go slowly.

Observe that $X$ is never equal to 1, nor 2. It might be equal to 3, to 4, ... does it stop somewhere? or can $X$ be very big ?
-> No, with 5 kids, there must be at least 3 of some gender. Indeed, the worst-case scenario is to have 2 girls and 2 boys after the 4th birth.
Conclusion: $X$ takes values in $\{3,4,5\}$.

You must compute separately, and by hand, $P(X=3), P(X=4), P(X=5)$. Take into account the symmetry Boy-Girl. and you can get through this. (check your answer with $P(X=3)+P(X=4)+P(X=5) =1$.

reemii · Answer

A start: $X=3$ means that the children born are either Boy-Boy-Boy or Girl-Girl-Girl. Therefore $P(X=3) = \frac12\frac12\frac12 + \frac12\frac12\frac12 = \frac14$.

Computing $P(X=4)$ is trickier: there are 3 Boys (or 3 Girls) but the 3rd Boy (or Girl) is the 4th child born. -> B-B-B-G and G-G-G-B are not possible situations.