"Background knowledge can even peg our probability estimate to an extreme: all the way to 0, or to 1. What’s Pr(U | C), the probability of an underage winner, given that he/she is a country singer? The intersection of U and C is zero, so this makes Pr(U | C) = 0. In words: a country winner eliminates any possibility of an underage winner. And what’s Pr(F | U), the probability that a woman wins, given that we know the winner to be underage? Well, F ∩ U and U are the same (check me), so Pr(F ∩ U) / Pr(U) = .4 / .4 = 1. Therefore, an underage winner guarantees a female winner.\n\nThe way I think about conditional probability is this: look at the diagram, consider the events known to have occurred, and then mentally block out everything except that. Once we know the background fact(s), we’re essentially dealing with a restricted world. Take the example of the known female winner. Once we know that event F in fact occurred, we can visually filter out David, and look at the F blob as though that were our entire world. In this restricted female-only view, the underage elements comprise a greater percentage of the total than they did before. And half of the rock-star elements have now been obscured, leaving only Kelly as the one-of-the-remaining-three.\n\nMany psychologists, by the way, claim that we’re constantly doing this sort of thing in our minds: gathering facts, then revising our beliefs about the world in light of those facts. We start by believing that Pr(X) is approximately some value. Then we learn K1 has occurred, and we update this to Pr(X | K1). Then we learn that K2 has also occurred, and so now we have Pr(X | K1 ∩ K2). (Can you see why it’s the intersection?) The more we learn, the more we revise our estimate up or down, presumably getting more accurate as we go. Another way of looking at it is that every time we learn something new is true, we also learn that its opposite is not true, and therefore we can eliminate some parts of the theoretically-possible universe that we have now ruled out. The denominator gets smaller and smaller as we eliminate possibilities."