Stats 101

It was spring or summer term. I was sitting in a classroom in a building adjacent to a grassy quad. Bikini-clad co-eds outside the window sunning themselves in the days before sun was considered unhealthy.

It was the first day of my first statistics class, and the professor opened with this story problem:
"A family moves in next door. You know they have 2 kids. You can see one of them playing in the yard-- it's a girl. What are the odds that the other is a boy?"

The answer wasn't 50-50. It also wasn't a trick question. Nothing about slightly more women in the population, or including potential effects of nuclear radiation on sexual identity.

The answer went like this:
With 2 kids, you have 4 possibilities.

Older	Younger
Boy	Boy
Boy	Girl
Girl	Boy
Girl	Girl

We see a girl, so the first option "boy-boy" is impossible. Out of the 3 remaining options, 2 out of 3 have a boy as the other child, so the answer is 2/3 or 67% chance that the other child is a boy.

Straightforward? Yep.

Clear? Absolutely.

Was I confident that I would be able to come to that conclusion by myself? Not at all. The fear I felt is likely the reason I still remember that story problem.

The trick is that when we see the girl playing in the yard we DON'T know if she is the older or younger child.

This leads to another interesting feature:

When we DON'T know if the girl is the older or younger child, we are 67% certain that the other child is a boy.

If we are told that, for example, the girl in the yard is the older child, then we are back to 50% certain that the other child is a boy. In this case, having MORE information makes us LESS certain.