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I'm struggling with the intuition here. I'm reading the Probability book by Blitzstein, where three situations are contrasted:

  1. There are two children, the eldest is a girl, what is the probability the youngest is also a girl? (1/2)
  2. There are two children, at least one is a girl, what is the probability the other is also a girl? (1/3)
  3. There are two children, you run into a random one on the street, and it's a girl, what is the probability the other is also a girl? (1/2)

Now, I understand the difference between situation 1 and situation 2: situation 1 has sample space {GG, GB, BG, BB}, and by knowing the eldest is a girl, this reduces the conditional universe to {GG, GB}. Situation 2 has the same sample space, but the conditional universe only reduces to {GG, GB, BG}. This all makes sense. However, I don't understand the difference between situation 2 and situation 3. Intuitively, my reasoning is that in both cases at least one child is a girl, and no information is given about whether it's the eldest or the youngest child, so they should have the same probability. Where is my reasoning mistaken?

user2296226
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2 Answers2

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I'm assuming that since it says "also a girl" in 3, it means you saw the child in the street is a girl. Given that, the other child could be boy or girl. It's the same as in situation 1, you know the exact status of a specific child, and it has no influence on the other child

Alan
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  • You are right, I edited my post to include that information. I still don't quite get the intuition behind the problem, but I understand the calculation so that's probably good enough. Thanks! – user2296226 Aug 14 '21 at 11:54
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    The intuition is that one coin flip doesn't affect another. So you meeting the one child takes that child out of the pool of possibility, so we are down to discussing is 1 unknown child male or female. – Alan Aug 14 '21 at 12:40
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For me the intuition works like this: In problem 2, you only know that there is at least 1 girl. Problem 3 however contains an extra random event: you meet a random child (once). Suppose you met a random child from this family 10 times but are blindfolded. Each time you are presented this random child and ask the parents if it's a boy or girl Each time they tell you it's a girl. After hearing 'girl' 10 times, wouldn't you be pretty confident that the family has 2 girls? This explains that even after meeting a girl once, as in problem 3, the probability is higher that the family has 2 girls, compared to that in problem 2.

Ed Jansen
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