What is the probability of this family having exactly four girls?

this is a tricky problem. and a simpler version of it has been debated much. see

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

there are 10 ways that out of five children 2 are girls 5!/(3!*2!)
10 ways that out of five children 3 are girls 5!(2!*3!)
5 ways that out of 5 children 4 are girls 5!(1!*4!)
1 way that out of 5 children 5 are girls

so the allowed space (at least two girls) has 26 possibilities

of these 5 are exactly 4 girls

so 5/26 is the answer

= Pr(X=4) / Pr( X >or = 2)

4/5

There are 32 different sex combinations for 5 children, e.g. GGGGG, GBGGG, BBGGG, BGBGB, etc.

6 of these combinations have only one girl or none so exclude those leaving 26 different combinations. Of these 5 have 4 girls and 1 boy so P(4G) = 5/26.

This assumes independence and P(B) = P(G) = 1/2 at each birth

As the gender of two of the children is know, let us ignore them and focus on the other three.

The following combinations are possible:
bbb, bbg, bgb, bgg, …, ggg
You have total of 2^3=8 combinations.
To have a total of exactly four girls, two of these children must be girls, the other a boy.
There are 3 different combinations (bgg, gbg, ggb) out of 8, so your chance to have exactly 4 girls in total is 3/8 = 37.5%