|Mathematicians don't understand people?
||[Dec. 31st, 2008|01:47 pm]
This is again a comment that I posted elsewhere, in response to the following question:
Someone tells you they have two children, and one of them is a girl. What are the odds that person has a boy and a girl?My answer, reproduced below, got voted up like crazy—I guess it struck a chord with nearly everyone. I know that many of you are math people.. I look forward to your vehement objections :-)
This is a famous puzzle. The answer is supposed to be 2/3, because what the question is asking you to do is consider all the parents in the world where at least one of the two children is a girl. Then you're left with 3 possibilities, BG, GB and GG.Seriously, when is the last time anyone told you that at least one of their kids was a girl? You could imply it, for instance by saying "I need to pick up my daughter from school," but I cannot think of a phrasing in English that lets you state it directly. Not without sounding like a huge dork, anyway.
If you phrase the question like that, everyone will get the right answer. The reason people get it wrong is that people don't normally talk like that. Imagine you're at a party, and someone tells you they have two kids, and "one of them is a girl." Clearly, they mean that the other is a boy, which means the answer is 100%.
But the most intuitive way of interpreting the question is that you know that a specific child is a girl, say because the person brought one kid to the party, who turns out to be a girl. With this interpretation, the obvious answer of 50% is in fact correct.
You often hear the complaint that people don't understand math. In this instance, however, an equally valid way of explaining what's going on is that mathematicians don't understand people.
This criticism applies partially to the normal game-show version of the Monty Hall problem, but I think there the wording is genuinely ambiguous regarding the host's behavior, and my answer would be "not enough information."
Edit. For the record, the missing information in the Monty Hall problem that I'm talking about is, "does the host only open doors which he knows not to have the prize, or any door other than the one the contestant picked?"