Is the following a binomial setting? A couple decides to continue to have children until their first girl is born; X is the total number of children the couple has.

Question

anonymous · Answer

Do you have an equation to go along?

anonymous · Answer

Nope, it's just yes or no

anonymous · Answer

Oh, then I would say yes.

anonymous · Answer

@AshMathNerd why is that?

anonymous · Answer

Because if you take X plus the girl it would be binomial.

anonymous · Answer

I think it would depend on the probability you want to find.

Say we want to find the probability that 8 girls are born out of 12 children. The probability of a girl being born would be $p=\dfrac{1}{2}$, and the probability of a boy being born would be $q=1-p=\dfrac{1}{2}$.

This question involves a binomial setting, yes. In fact, the probability would be
$$\dbinom{12}{8}p^{8}q^{12-8}=\dbinom{12}8\frac{1}{2^{12}}\approx0.121 $$
However, let's say we wanted to find the probability that the $n$th child is born a girl. Now this involves a geometric probability, which would be given by the formula $q^{n-1}\times p$.

So as an example, the probability that the 3rd child is born a girl would be
$$\left(\frac{1}{2}\right)^{3-1}\frac{1}{2}=\frac{1}{2^3}=\frac{1}{8}=0.125$$
To me, the question is a bit vague, but that could be a misunderstanding on my part.