A couple days ago, Jeff Atwood posted the following puzzle:
Let’s say, hypothetically speaking, you met someone who told you they had two children, and one of them is a girl. What are the odds that person has a boy and a girl?
Consider your answer carefully, without doing a web search, or reading the comments to this post. Don’t cheat — but be prepared to explain your reasoning, because the solution might surprise you.
The 800+ message comment thread that ensued is an interesting study in how people go about resolving cognitive dissonance. I dare not even look at the other 800+ message thread from when Atwood posted the solution.
The problem is interesting, because it forces the reader to overcome two separate intuitive hurdles. The first is that past events do not impact the probability of future events. If I flip a fair coin nine times and the first nine times are heads, what are the odds that the tenth will also be heads? 50% — the first nine flips have no impact on the probability of the tenth flip.
So the typical reader has been “primed” to read this problem and think, “Aha! — you can’t fool me! The sex of the first child has absolutely no impact on the sex of the second child. The answer is 50%!”
But the question is about the aggregate outcome, not an individual outcome. And the aggregate outcome of one boy, one girl is twice as likely as the aggregate outcome of two girls, so the answer is 2/3.
This is a puzzle, not a trick question. But when presented with a puzzle like this, we tend to think it is a trick question, that we are being given extraneous information that is a red herring, and hunt for the flash of insight that will give us a shortcut to solving the problem with little further effort.
Except in this case, the flash of insight is the red herring. If we allow it to blind us, and don’t actually think through the math of the situation, lay out the possibilities, etc., we’ll end up with the wrong answer.
I submit that the real world has a lot more problems like this, that are solved by old-fashioned elbow grease than by a flash of insight. But there’s enough problems that are solved by a flash of insight that we sometimes feel stupid when we pick up our pencils and start grinding through the math. But that’s usually what it takes.
