Dear Uncle Colin,

I have a probability question that involves a weird place where every family has three children, and every child is either a girl or a boy. 50-50. Independent of each other. If I take a random family, and choose at random one of the children, I have to find the probability that this child has a brother and a sister given that I picked a girl. I’ve come up with several methods, but they all give different answers!

- Families! And Mine Isn’t Like Yours!

Hi, FAMILY, and thanks for your message!

I have two possible methods for you here: a brutal one and an elegant one.

Brutally

There are eight possible families:

  • GGG
  • GGB
  • GBG
  • GBB
  • BGG
  • BGB
  • BBG
  • BBB

Each of the 24 children is equally likely to be picked. Out of them, 12 are girls; six are girls with a brother and a sister. The probability of that being the family arrangement, given you’ve picked a girl, is $\frac{6}{24} \div \frac{12}{24} = \frac{1}{2}$.

Elegantly

Each child could have one of four possible orderings of siblings: BB, BG, GB or GG.

Because all of the probabilities are 50-50 and independent, the probability of any child having a brother and a sister is two in four, or $\frac{1}{2}$ again.

Hope that helps!

- Uncle Colin