3 ms·

I'm confused about why people argue that it is not necessarily 2/3. I assume that the sample space is: [(b,b),(b,g),(g,b),(g,g)] where the four possible birth

by asorbus 17y ago

I'm confused about why people argue that it is not necessarily 2/3. I assume that the sample space is: [(b,b),(b,g),(g,b),(g,g)] where the four possible birth sequences are [(younger child is boy, older child is boy), (younger child is boy, older child is girl), (younger child is girl, older child is boy), (younger child is girl, older child is girl)] and the sequences are all equally likely. So now if you ask, what is the probability that the person has a girl and a boy given that at least one of them is a girl, you can discard (b,b) from your sample space because you know that there is at least one girl. Now your sample space is [(b,g),(g,b),(g,g)] and the odds that the person has a girl and a boy are 2/3.

anatoly 17y ago
The way you phrase it - "given that one of them is a girl" - the answer is 2/3. The people who argue that it is not 2/3 and yet aren't confused are choosing a different interpretation of the problem. When your acquaintance tells you "one of my children is a girl", they interpret it not as pure information, so to speak, but as an event to condition on. It's reasonable to suppose, for example, that your acquaintance could have said "one of my children is a girl" or "one of my children is a boy", and used the actual knowledge of his children's gender to pick one of the two possibilities to say (choosing randomly if it's a boy or a girl). In that case, the question becomes: given that he/she said "... a girl" rather than "... a boy", what is the probability that it's a girl and a boy? In that case, the sample space becomes larger: to (b,b) etc. add also B or G, which is whether the acquaintance said "one of my children is a boy" or "... a girl". The possibilities are: b,b,B: 1/4 (b,b,G is impossible) g,g,G: 1/4 (g,g,B is impossible) b,g,B: 1/8 b,g,G: 1/8 g,b,B: 1/8 g,b,G: 1/8 Now if you condition this sample space on the event "G", you rule out exactly half the possibilities, and the answer is 1/2.

by asorbus 17y ago

anatoly 17y ago

The way you phrase it - "given that one of them is a girl" - the answer is 2/3. The people who argue that it is not 2/3 and yet aren't confused are choosing a different interpretation of the problem. When your acquaintance tells you "one of my children is a girl", they interpret it not as pure information, so to speak, but as an event to condition on. It's reasonable to suppose, for example, that your acquaintance could have said "one of my children is a girl" or "one of my children is a boy", and used the actual knowledge of his children's gender to pick one of the two possibilities to say (choosing randomly if it's a boy or a girl). In that case, the question becomes: given that he/she said "... a girl" rather than "... a boy", what is the probability that it's a girl and a boy? In that case, the sample space becomes larger: to (b,b) etc. add also B or G, which is whether the acquaintance said "one of my children is a boy" or "... a girl". The possibilities are: b,b,B: 1/4 (b,b,G is impossible) g,g,G: 1/4 (g,g,B is impossible) b,g,B: 1/8 b,g,G: 1/8 g,b,B: 1/8 g,b,G: 1/8 Now if you condition this sample space on the event "G", you rule out exactly half the possibilities, and the answer is 1/2.