Comment by tyn | Hacker News Reader

Comment by tyn | Hacker News Reader

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From the Jeffrey Rosenthal paper that Jeff Atwood suggests (http://www.probability.ca/jeff/writing/montyfall.pdf):

<Quote>

Nebulous Neighbours:

Your new neighbours have two children of unknown gender. From older to younger, they are equally likely to be girl-girl, girl-boy, boy-girl, or boy- boy. One day you catch a glimpse of a child through their window, and you see that it is a girl. What is the probability that their other child is also a girl?

Solution.

The probabilities that a glimpsed child will be a girl for each of the four possi- bilities (girl-girl, girl-boy, boy-girl, and boy-boy) are respectively 1, 1/2, 1/2, and 0. Since the probabilities must add to 1, the probabilities of these four possibilities are respectively 1/2, 1/4, 1/4, and 0. Hence, the probability is 1/2 that the other child is also a girl.

</Quote>

Can someone explain how glimpsing to one child and seeing that is a girl is different from hearing the parent saying that one child is a girl? How come and getting the same information from your eyes instead of your ears changes the probabilities? (1/2 for the other child to be a boy when seeing, 2/3 when hearing).

Seeing vs hearing is a red herring.

What really matters is whether information about one child being a girl fixes a particular child. It doesn't matter how the child is fixed (a popular variation is "I have two chilren and the older one is a girl"), but it does matter that out of the two children, it is now known about a specific one of them that it is a girl. In that case, out of 4 possibilities, 2 are ruled out, and the answer is 1/2.

If all that is known is that one of the children is a girl, only one possibility out of 4 is ruled out, and the answer is 2/3.

"If all that is known is that one of the children is a girl, only one possibility out of 4 is ruled out, and the answer is 2/3."

Well, that's the case when you glimpse from the window. Probably I'm wrong but I don't see why there is a fix, you still don't know if you see a G in GB or in BG or in GG

The difference is in how the information got revealed (whether it was seen or heard is irrelevant). In the case where you saw the girl, it completely specifies the gender of the child you saw and unspecifies the gender of the unseen child. (Intuitively: suppose you know the family has four children, and you peek in the window and see that three of them are girls. What's the probability of the fourth being a girl? 1/2.)

The case where you are told the family has at least one girl is potentially different... It depends on how you get this information. If you ask the parent, "Do you have at least one girl?" and they say yes, then the probability of the other child being a girl is 1/3. On the other hand, if you ask the parent, "What can you tell me about your children?" and they say "I have at least one girl", then the probability of the other child being a girl is 1/2. The difference is that in the second case, the parent chooses which child to reveal information about, while in the first case, they are forced to make a combined statement about both children. In the second case, the parent completely reveals the gender of the chosen child while unspecifying the gender of the other, just as good as seeing the chosen child in the window.

Obviously, most people who describe the boy-girl problem are interested in describing the non-intuitive 1/3 result. But in their rush to reveal the "paradox", many state the problem poorly, leaving open the possibility of the 1/2 case (as Jeff did when he presented the problem).

(I added a similar response with more details in the Wikipedia article discussion several months ago: http://en.wikipedia.org/w/index.php?title=Talk:Boy_or_Girl_p... I must warn, however, that the concluding paragraph there is incorrect: Jeff's problem is fundamentally undecidable as stated.)

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