Well, an interesting set of answers.
Here is the correct one:
2/3 or .667 -- the probability that the family also has a boy is 2/3. (cheers jfc)
And here's the logic:
You were told a family has two children. There are four possible ways in which a family can have two children:
#1 Girl-Girl (probability = .25)
#2 Girl-Boy (prob = .25)
#3 Boy-Girl (prob = .25)
#4 Boy-Boy (prob = .25)
Each of those four ways of having two children has an equal probability of happening, and those probabilities, of course, add up to 1.000. (as specified in the question: Assuming that the biological probability of having either a boy or a girl baby is equal 50-50).
You were told that one of the children was a girl. That only eliminates one possibility out of the four: Boy-Boy, leaving three other possibilities, (GG, GB, BG) all of equal probability. Of those three, two include a boy. Thus the probability that the family ALSO has a boy is 2/3 or .667.
The most common mistake that people make when confronted with this problem is that they try to reduce it to a simpler problem. The mistake is in thinking that the FIRST child was a girl, so what is the probability that the SECOND child is a boy. In that improperly simplified problem, you have eliminated TWO out of the four possible ways of having two children (Boy-Girl and Boy-Boy) leaving only two possibiliities, only one of which has a boy in it. And thus the mistaken 50% answer. The mistake was in eliminating Boy-Girl from the set of possible situations during the simplification.

thank you