The "second-sibling" problem

The "second-sibling " problem has spurred a lot of discussion.

The problem (slightly rephrased to avoid confusion):
"Let's say, hypothetically speaking, you met someone who told you that some old friends of yours have two children, and one of them is a girl. What are the odds that they have both a boy and a girl?"

Original probability sets:

1 child:
A boy or girl: 50% probability

2 children:

If you know that one of the children is a girl, you must remove the first option (1 boy + 1 boy) as that is impossible - it can be a boy in 2 (B and C) of the 3 remaining cases (2/3), or in (25%+25%) / (25%+25%+25%) = 50% / 75% (= 2/3)

The problem here is if you change the question and say that you know that the first child is a girl, because then you remove option A and B as they can't be possible. And then you're left with 2 options which gives 50% probability.

Most people find this distinction difficult to understand...

A different example:

A couple wants two children, and they select two names each, as they don't know if they'll get girls or boys (first name preferred)

If boys: brad and tom
If girls: angelina and katie

Original possible outcomes

You hear from some relatives that they got two children, and one of the children is a girl (named Angelina). What are the odds that they have both a boy and a girl?

It cannot be Brad and Tom, as one of the children is a girl. Therefore it is only 3 possible outcomes left.

2 out of 3 possible outcomes is a boy and a girl (B and C)
1 out of 3 possible outcomes is two girls (D)
Two boys is impossible (A)

If they say that their first child was a girl (Angelina), then you have

and you have a 50% chance (2 out of 4) of them having both a boy and a girl.

So, the main difference is that in the original question, we know that one of the children is a girl, but we don't know which (the first or the second). Therefore, we can only eliminate 1 of the 4 possibilities (Boy+Boy). If we know they got a girl first, we can eliminate 2 possibilites (boy+boy and boy+girl).

Also remember that we are not considering what gender the next child is, but the probability based on both children given that we also know the gender of one of the children.

Links:
- http://en.wikipedia.org/wiki/Marilyn_vos_Savant#.22Two_boys.22_problem