Probability of Two Girls: Understanding Conditional Probability

a) The probability that both children are girls given that the oldest is a girl is 1/2. This is because the gender of the oldest child does not affect the gender of the youngest child, so each child still has a 1/2 probability of being a girl.

b) The probability that both children are girls given that one of them is a girl is 1/3. If we denote the children as A and B, and assume that the oldest child is referred to as the 'one of them' mentioned, there are four equally likely possibilities: AG, GA, GG, and GB (where G represents a girl and B represents a boy). Out of these possibilities, only GG represents both children being girls. Thus, the probability is 1 out of 3.