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Two Children

1) Mr. Jones has two children.
The older child is a girl.
What is the probability that both children are girls?"

2) Mr. Smith has two children.
At least one of them is a boy.
What is the probability that both children are boys?

1) There are four equally probable events GG,GB,BG,BB.
Only two of these possible events meets the criteria specified in the question (e.g., GB, GG).
Since both of the two possibilities in the new sample space {GB, GG} are equally likely, and only one of the two, GG, includes two girls, the probability that the younger child is also a girl is 1/2.

2) Again, there are four equally probable events GG,GB,BG,BB.
If it is assumed that both children were considered, the answer is 1/3.
However, if it is assumed that the information was obtained by considering only one child, then the answer is 1/2.