Comment by pontus | Hacker News Reader

Comment by pontus | Hacker News Reader

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Just to pile on here, there's also ambiguity around how the observed girl is selected. Consider the following framing:

I go to a random house on a random street and knock on the door. A young girl opens the door. I ask how many siblings they have and they say one. What's the probability that they have a sister?

Now it's 50% even though cosmetically it seems like it'd be fair to say that the family has at least one daughter. The reason is that once I see a girl at the door, I'm slightly more confident in that it's a GG household since a GB or BG household would sometimes show a boy opening the door (assuming the two kids are equally likely to open the door).

P(GG | G at door) = P(G at door | GG) P(GG) / P(G at door)

P(G at door) = 1/2 (by symmetry)

So, P(GG | G at door) = 1 * 1/4 * 2 = 1/2

This is the crux of the "paradox," which is really just an interpretation problem. Most people assume that the question asks exactly your scenario, i.e. if a specific child is selected and it's a girl, what's the probability that the sibling is also a girl? In that case, the event space is just GB or GG, and p(GG)/(p(GB) + p(GG)) = 0.5. (BG is not in the event space because we are conditioning on a specific child being a girl.)

However, if the question is interpreted as "what's the probability of having two girls if we know there aren't two boys," then the event space is GB, BG, GG, and p(GG)/(p(GB) + p(BG) + p(GG)) = 1/3. Both GB and BG are in the event space because we are not conditioning on the sex of one specific child.

You’re making the classic mistake in conflating computing how pathways to conditions can rise from computing conditionals given the current state. There’s absolutely no information theoretical difference between you saying “A girl opened the door” and “I was told the family has a girl.”

Look at the more technical descriptions using conditional probabilities of the Monty Hall problem as it is essentially equivalent. You’re trying to factor in the probability of whether Monty knows if a goat is behind a door when the observable information is that there is an open door with a goat. One you make that observation many things collapse.

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