Question

For a family with 2 children, the sample space indicating boy (B) or girl (G) is BB, BG, GB, and GG. If each of the outcomes is equally likely, find the probability that the family has 2 girls, given that the first child is a girl.

Answer 1

This is a problem of conditional probability. The shortcut to the answer is consider only the outcomes where the first child is a girl, and calculate the probability among those where both are girls.
I.E.
Ω={GG,GB}
favourable outcome: {GG}
So assuming all outcomes are equally probable, then
P(GG|first child is a girl)=1/2

The proper way is to use the definition:
P(GG|GX) ...both girls given first child is a girl
=P(GG∩GX)/P(GX)
=P(GG)/P(GX) .. since P(GG&GX)=P(GG) by absorption
=(1/4)/(1/2)
=1/2