Innumeracy: Mathematical Illiteracy and Its Consequences

by John Allen Paulos

Imagine a man with three cards. One is black on both sides, one red on both sides, and one black on one side and red on the other. He drops the cards into a hat and asks you to pick one, but only to look at one side; let’s assume it’s red. The man notes that the card you picked couldn’t possibly be the card that was black on both sides, and therefore it must be one of the other two cards—the red-red card or the red-black card. He offers to bet you even money that it is the red-red card. Is this a fair bet? At first glance, it seems so. There are two cards it could be; he’s betting on one, and you’re betting on the other. But the rub is that there are two ways he can win and only one way you can win. The visible side of the card you picked could be the red side of the red-black card, in which case you win, or it could be one side of the red-red card, in which case he wins, or it could be the other side of the red-red card, in which case he also wins. His chances of winning are thus 2⁄3. The conditional probability of the card being red-red given that it’s not black-black is ½, but that’s not the situation here. We know more than just that the card is not black-black; we also know a red side is showing.