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.
Example 19.1 The Monte Hall Problem
Monte Hall was the flamboyant host of a television game show created in the 1960s called Let's Make a Deal. Contestants on the show were led by Hall through a series of guessing games and deals for cash and prizes. The scenario for our problem is the following: the prize is a new car, which is hidden behind one of three identical doors. Behind the other two doors are cows. The contestant must guess the correct door to win the car. Once the contestant has chosen a door, Monte Hall opens one of the remaining doors, and a cow walks through onto the stage.[…]
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