1) If a man has two children, one of whom is a boy, what is the probability that both children are boys?
Most people guess that the answer is one-half: you know that one child is a boy, and you assume that the other is equally likely to be a boy or a girl.
This is incorrect. Since the boy could be the older or the younger child, the possible children and their orderings are boy then girl, girl then boy, or boy then boy. So the probably of both children being boys is not one-half, but rather one-third.
2) If a man has two children, one of whom is a boy born on Tuesday, what is the probability that both children are boys?
After reading and understanding the answer to question 1, you're probably thinking: Tuesday has nothing to do with this, it's just a distraction. The answer is still one-third.
But this is wrong. Now you have to list out the possibilities of births according to gender and days of the week. If the older child is a boy born on Tuesday, there are fourteen possibilities for the traits of the second child: girl or boy born on each of seven days. But if the younger child is the Tuesday-born boy, there are only thirteen unique possibilities for the traits of the older child, since we've already counted the situation of two boys both being born on a Tuesday. Out of these 27 total possibilities, two boys account for 13 of them. Thus the probability is 13/27, or 48%.
So you see that knowing what day the boy was born on has changed the game by making it significantly more likely that the other child is a boy. As you learn more information, the probability of two boys gradually approaches one-half. This has to do with the fact that the boys become distinguishable.
These questions confuse the hell out of everyone who thinks about them; even mathematicians have been up in arms since Martin Gardner posed them in 1959. Controversy remains as to how the wording of the problems affects the answers. For example, the answer to the first question relies on the fact that the father-of-two was selected at random from amongst fathers of two, not from amongst fathers-of-two-with-at-least-one-boy. It's a really subtle point, and you can read about it here.