Some Probability Theory The math department’s undergraduate course in probability theory is MATH 411, Mathematical Probability. Thefirst person to attempt a systematic theory of probability was the Italian mathematician, physician, astrologer, and gambler Gerolamo Cardano (1501–1576). Cardano is perhaps best known for his study of cubic and quartic algebraic equations, which he solved in his 1545 text Ars Magna (The Great Art)—solutions which required the controversial notion of keeping track of the unlikely expression √ 1. − 1 In 1654, Antoine Gombaud Chevalier de Mere (1607–1684), a French nobleman, writer and professional gambler, brought Blaise Pascal’s attention to the following game of chance: is it worthwhile betting even money that double sixes will turn up at least once in 24 throws of a fair pair of dice? This led to a long correspondence between the French mathemati- cians Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665), during which the classical theory of probability wasfirst formu- lated. Pascal is on the left. 2 The single most influential work on probability theory was writ- ten by another French mathematician Pierre Simon Laplace (1749– 1827), “Thèorie Analytiques des Probabilitès,” published in 1812. As a starting point for our discussion of probability theory, let’s see if we can answer de Mere’s question: is it worthwhile betting even money that double sixes will turn up at least once in 24 throws of a fair pair of dice? 3 4 5 Roulette The game of American roulette employees a large wheel with 38 slots, 36 labeled non-sequentially with the numbers 1–36, and the two house slots labeled 0 and 00. Exactly half of the 36 numbered slots are red, while the other half are black. The house slots are typically shaded green. There are a number of different ways to bet in Roulette. The simplest is on red or black: the player (Rose) bets $1.00 on one or the other. If she wins, she gains $1.00, and if she loses, she loses $1.00 (even money). It’s natural to ask the following question: what is Rose’s expected return on a bet of $1.00 on red or black? 6 7 Random Variables and Expected Value We often describe the outcome for a game of chance such as roulette with a random variableX. 8 Doubling Up One well-known gambling strategy that can be used with just about any game is the idea of “doubling up.” The idea is simple: Rose begins by betting $1.00 on red or black. If she wins, she takes her dollar and starts over, clearly having earned exactly $1.00. If she loses, she doubles her bet to $2.00. If she wins the second play, she takes her two dollars and starts over, again having earned exactly $1.00. If she loses the second play, she doubles her bet again to $4.00. If she wins the third play, she takes her four dollars and starts over, once again having earned exactly $1.00. If she loses the third play, she doubles yet again. Continuing this way, and assuming she will always win eventually, Rose ends every full round of plays be earning precisely $1.00. Let’s ask the following question: what happens if a limit is placed on the amount Rose can bet? For example, suppose the casino Rose frequents has a betting limit of $4.00. What is her expected value betting on red or black for a full cycle of play, and how does it compare to the expected amount bet? 9 10 The St. Petersburg Paradox As afinal example, we’ll talk about the St. Petersburg Paradox, which was suggested by the brothers Daniel and Nicolaus Bernoulli in 1725. Daniel is on the left. Here’s the game: a “dealer”flips a fair coin until it lands heads. IfN denotes the number offlips required, then he pays the player 2N . If heads turns up on thefirstflip, the player get $2.00. If heads doesn’t turn up until the secondflip, the player gets $4.00 and so on. The Bernoulli Brothers asked the following question: how much would you be willing to pay to play this game? 11 12.