A gambler may not know, off the top of his or her head, the exact probability of the winning cards in a hand of Poker or a doctor the exact likelihood of a particular diagnosis. They both may have a gut feeling for how likely or unlikely such an event is to happen. Calculating a probability provides, in this sense, a precise quantification for our intuition about the likelihood of an uncertain event.

In this chapter, we pursue a class of examples designed to stretch intuition and test beliefs about the idea of probability. Many of the examples presented here and in the exercises are commonly referred to as paradoxes. A paradox is a statement or problem leading to contradictory conclusions or solutions. True paradoxes in mathematics are rare and significant phenomena. For example, a simple paradox about “truth” known as the Liar Paradox (see Exercise 10.6) ultimately led to Gödel's remarkable Incompleteness Theorem in mathematical logic. In our own study, the Condorcet Paradox is the basis for Arrow's Impossibility Theorem (Theorem 18.19) and the central example in social choice theory. While we retain the term “paradox” in the names when this is common use, the question of which examples to follow are truly paradoxical is an interesting one and left to your interpretation. We begin with what is perhaps the most popular example of a probability puzzle.