# Probability of having exactly $1$ girl amongst $n$ kids?

I have $$n$$ kids and I want to determine the probability that I will have at most, $$1$$ girl. What is the probability of this happening?

We know that:

$$P(\text{at most 1 girl}) = P(\text{no girls}) + P(\text{exactly 1 girl})$$

For $$n$$ many kids, I know that $$P(\text{no girls})=\frac{1}{2^n}$$, because every other combination will have at least $$1$$ girls, however how can I determine $$P(\text{exactly 1 girl})$$?

• This is a binomial distribution – David Lui May 4 at 2:33
• We haven't covered that so we are not allowed to use it. – scarface May 4 at 2:35

The answer is essentially a sneaky use of the binomial. You have $$n$$ children $$n-1$$ boys $$Prob=\frac{1}{2^{n-1}}$$ plus one girl $$Prob=\frac{1}{2}$$. However in the sequence of children the girl could be in any spot so the probability of exactly one girl is $$\frac{n}{2^n}$$.