Note: This corrects a misprint in my previous article which makes *me* look really stupid. I hope the previous version has been cancelled.
In article <firstname.lastname@example.org>, Eldon Moritz <email@example.com> wrote: >> From: Alan Morgan <amorgan@CS.Stanford.EDU> >> In article <firstname.lastname@example.org> Eldon Moritz writes: >> > >SNIP< >> >"A lady has two children, at least one of which is a girl. What is the >> >probability that she has two girls?" >> > >> >The answer is one-half, but the conventional argument seems to say it is >> >one-third. Very obviously there is a flaw, an error in the conventional >> >argument. Can you find it? >> >> Nope. There isn't one (unless you are putting some creative spin on >> the problem that other people are not). > >Here we are opposed. I said there is a fallacy. You said flat out there is >not. You think I am lying or real stupid or probably both.
It is very unlikely that you are lying. And you don't have to be stupid to make the mistake you did. In fact, it's the same mistake almost everyone makes when they first encounter this problem. And in fact, it's not even clear that it is a mistake, since it's mostly a question of how you interpret the problem.
It's very difficult to state this problem in a way that makes it obvious what is given in the problem. The trouble is that most people don't realize that there are two possible interpretations. Once you realize that, I think most people would agree that the phrase "at least one" should be interpreted in the way it is in probability courses. This interpretation makes you wrong.
First possibility: You encounter a lady on the street. She has a little girl with her who is her daughter. You ask her if she has any other children and she has, "Yes, I have another child."
What is the probability that her second child is a girl? One-half, just as you have claimed.
Second possibility: You are in the audience for a talk show. The host asks all the women in the audience who have at least one daughter to raise their hands. He chooses one of the women who have raised their hands and asks, "How many children do you have?" She answers "Two." What is the probability that both her children are girls? Answer: One third, just as is claimed in probability classes.
Now I realize that for many people, it's still hard to see at first why these two situations are different. So consider a third possibility: Same audience, host asks all the women in the audience whose *youngest* child is a girl to raise their hands. He chooses one and asks, "How many children do you have?" and she answers "Two." What is the probability that the other child is also a girl? Answer: One-half, just as you claim.
(Fourth possibility: Instead of saying "youngest" he says "oldest." The answer is still one-half.)
The third situation is essentially the same as the first possibility. The question of whether you're really stupid or not can be settled by seeing whether, after thinking about it, you can understand why this is different from the second situation.
Can you see that in the third situation there will be fewer women with their hands raised than in the second? Can you also notice that all the women who have two daughters will have their hands raised both times (assuming that all the women in the audience have exactly two children)? Can you see why this means that the proportion (probability) is going to be different in the two cases?
-- Trying to understand learning by studying schooling is rather like trying to understand sexuality by studying bordellos. -- Mary Catherine Bateson, Peripheral Visions lady@Hawaii.Edu