From John Allen Paulos:
Consider the two boys problem in probability. Given that a family has two children and that at least one of them is a boy, what is the probability that both children are boys? The most common solution notes that there are four equally likely possibilities — BB, BG, GB, GG, the order of the letters indicating birth order. Since we’re told that the family has at least one boy, the GG possibility is eliminated and only one of the remaining three equally likely possibilities is a family with two boys. Thus the probability of two boys in the family is 1/3. But how do we come to think that, learn that, believe that the family has at least one boy? What if instead of being told that the family has at least one boy, we meet the parents who introduce us to their son? Then there are only two equally like possibilities — the other child is a girl or the other child is a boy, and so the probability of two boys is 1/2.
Many probability problems and statistical surveys are sensitive to their intensional contexts (the phrasing and ordering of questions, for example). Consider this relatively new variant of the two boys problem. A couple has two children and we’re told that at least one of them is a boy born on a Tuesday. What is the probability the couple has two boys? Believe it or not, the Tuesday is important, and the answer is 13/27. If we discover the Tuesday birth in slightly different intensional contexts, however, the answer could be 1/3 or 1/2.