Answer:

In a family with 2 CHILDREN there are four possibilities:

1) the first child is a boy and the SECOND child is a boy (bb)

2) the first child is a boy and the second child is a girl (bg)

3) the first child is a girl and the second child is a boy (gb)

4) the first child is a girl and the second child is a girl (gg)

Since we are given that at least one child is a girl there are THREE possibilities: bg, gb, or gg. Out of those three possibilities the only one with two girls is gg. Hence the probability is 13.

Step-by-step explanation:

I think this question CONFUSES a lot of people because there's a lack of intuitive context -- I'll try to supply that.

Suppose there is a birthday party to which all of the girls (and none of the boys) in a small town are invited. If you run into a mother who has dropped off a kid at this birthday party and who has two children, the chance that she has two girls is 1/3. Why? 3/4 of the mothers with two children will have a DAUGHTER at the birthday party, the ones with two girls (1/4 of the total mothers with two children) and the ones with one girl and one boy (1/2 of the total mothers with two children). Out of these 3/4 of the mothers, 1/3 have two girls.

On the other hand, if the birthday party is only for fifth-grade girls, you get a different answer. Assuming there are no siblings who are both in the fifth grade, the answer in this case is 1/2. The child in fifth grade is a girl, but the other child has probability 1/2 of being a girl. Situations of this kind arise in real life much more commonly than situations of the other kind, so the answer of 1/3 is quite nonintuitive.