A man and a woman (unrelated) each have exactly two children. At least one of the man's children is a boy, and the woman's older
child is a boy. You're going to guess that each of them has two boys. What is the probability that you're right in each case?
You can assume that the probabilities of having a boy or girl are equal.

I don't know too much about statistics but since no one has answered it yet I'll tack a crack. So they have two children each and in each case there is at least one boy. What are the chances of both other children being boys. Well in each case there is a 1/2 chance of the other sibling being a boy. So, the chance of both families having two boys would be 1/2+1/2=1/4. I can't remember if you add or multiply them but in this case it wouldn't matter.

Let's start with the woman, because that's an easier case to understand. If the older child is a boy, what is the probability that the younger is a boy? There are two possibilities - the younger could be either a boy or a girl, so the probability of the younger being a boy is 1/2.

Now consider the man. You know he has at least one boy, but you don't know whether the boy is the older child or the younger. The birth order could be any one of these:

Boy, Girl
Boy, Boy
Girl, Boy

Hence the probability of the man having two boys is 1/3.

It's unclear whether the question is asking for the probability of both the man and the woman having two boys, but to find this you multiply the two probabilities since they are independent events:

Prob(man has 2 boys AND woman has 2 boys) = 1/3 x 1/2 = 1/6.