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 $P(\text{exactly 1 girl})$?

  • $\begingroup$ This is a binomial distribution $\endgroup$ – David Lui May 4 at 2:33
  • $\begingroup$ We haven't covered that so we are not allowed to use it. $\endgroup$ – 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 $\frac{n}{2^n}$.

  • $\begingroup$ Is there a more general formula for this? $\endgroup$ – scarface May 4 at 3:32
  • $\begingroup$ General formula is based on binomial distribution. $\endgroup$ – herb steinberg May 4 at 3:52

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