Well-known, but don't look up

[iago]

I think the "random" part is important. If you just said, "a lady has two children, one is a son. What are the odds that the other is a son", I think it's more ambiguous. If you went out and found a lady that had a son, then the odds that the other is a son would be, I think, 50%. But if you found a random person, determined that she had one son, and asked what the other might be, that's different.

I think. :)

[Nate]

Answer is .33

Order isn't important so there are 3 different states, BB, GG, BG
Each has the same probability P, the sum of the probabilities has to equal 1,
so 3P = 1 or P = .33.

[Rule]

I think the "random" part is important. If you just said, "a lady has two children, one is a son. What are the odds that the other is a son", I think it's more ambiguous. If you went out and found a lady that had a son, then the odds that the other is a son would be, I think, 50%. But if you found a random person, determined that she had one son, and asked what the other might be, that's different.

I think. :)

No.  If you only sought out a lady who had a son and one other child, and you know nothing else about the lady, then there is no difference.

However, if a lady has one son, and she is pregnant, the probability that the child-to-be is a boy is 50%. It seems, superficially, that the situations are identical.  After all, the lady will eventually have one boy, and one other child.  But giving birth shouldn't change the probability that the other child is a boy.

I'll pose this as a new problem :P.  Explain the difference that accounts for the difference in probability.

[Rule]

Answer is .33

Order isn't important so there are 3 different states, BB, GG, BG
Each has the same probability P, the sum of the probabilities has to equal 1,
so 3P = 1 or P = .33.

? She has one boy.  GG is not a possible state.  You also have to justify why the three actual possible states have equal probability.

[Rule]

Explain the difference that accounts for the difference in probability.

Somehow I think there will be no takers.  lol

[rabbit]

You guys seem to be forgetting that there's also the possibility of hermaphrodites, identical twins, and fraternal twins.