"A family has two children, at least one of whom is a boy; what is the probability that both are boys?"
When I teach this material, I tell the students that one standard version of the question -- "A family has two children, one of whom is a boy; what is the probability that the other one is a boy?" -- is too incoherent to have a clear answer. The first half of the question seems to be saying "at least one of the children is a boy", but if so, then the second half commits the fallacy of misplaced concreteness by talking about a boy in particular. Another instance of misplaced concreteness is the fiction of an average person, an average family, etc. One can devise a situation in which the average family has four children, but the average child has five siblings! (Posit a three-family house: two of the families have 2 children each, and the third family has 8 children.) Or, say I have two cars, one of which gets 10 miles to the gallon and the other of which gets 40 miles to the gallon. Then my "average car" goes 25 miles on a gallon of gas, but uses 1/16 of a gallon per mile! Jim Propp