The Best Writing on Mathematics 2013

Mircea Pitici, Mircea Pitici

• Publisher: Princeton University Press
• Year: 2014
• ISBN:
  9781400847990
• Topic: Mathematics
• Subtopic: Mathematics Essays
• Index: Mathematics

The year's finest writing on mathematics from around the world, with a foreword by Nobel Prize – winning physicist Roger Penrose This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2013 makes available to a wide audience many articles not easily found anywhere else—and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Philip Davis offers a panoramic view of mathematics in contemporary society; Terence Tao discusses aspects of universal mathematical laws in complex systems; Ian Stewart explains how in mathematics everything arises out of nothing; Erin Maloney and Sian Beilock consider the mathematical anxiety experienced by many students and suggest effective remedies; Elie Ayache argues that exchange prices reached in open market transactions transcend the common notion of probability; and much, much more.In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed mathematical physicist Roger Penrose and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us—and where it is headed.

Errors of Probability in Historical Context

PRAKASH GORROOCHURN

1. Introduction

This article outlines some of the mistakes made in the calculus of probability, especially when the discipline was being developed. Such is the character of the doctrine of chances that simple-looking problems can deceive even the sharpest minds. In his celebrated Essai Philosophique sur les Probabilités (Laplace 1814, p. 273), the eminent French mathematician Pierre-Simon Laplace (1749–1827) said,

… the theory of probabilities is at bottom only common sense reduced to calculus.

There is no doubt that Laplace was right, but the fact remains that blunders and fallacies persist even today in the field of probability, often when “common sense” is applied to problems.

The errors I describe here can be broken down into three main categories: (i) use of “reasoning on the mean” (ROTM), (ii) incorrect enumeration of sample points, and (iii) confusion regarding the use of statistical independence.

2. Use of “Reasoning on the Mean” (ROTM)

In the history of probability, the physician and mathematician Gerolamo Cardano (1501–1575) was among the first to attempt a systematic study of the calculus of probabilities. Like those of his contemporaries, Cardano’s studies were primarily driven by games of chance. Concerning his 25 years of gambling, he famously said in his autobiography (Cardano 1935, p. 146),

… and I do not mean to say only from time to time during those years, but I am ashamed to say it, everyday.

Cardano’s works on probability were published posthumously in the famous 15-page Liber de Ludo Aleae,¹ consisting of 32 small chapters (Cardano 1564). Cardano was undoubtedly a great mathematician of his time but stumbled on several questions, and this one in particular: “How many throws of a fair die do we need in order to have a fair chance of at least one six?” In this case, he thought the number of throws should be three.² In Chapter 9 of his book, Cardano says of a die:

One-half of the total number of faces always represents equality³; thus the chances are equal that a given point will turn up in three throws …

Cardano’s mistake stems from a prevalent general confusion between the concepts of probability and expectation. Let us dig deeper into Cardano’s reasoning. In the De Ludo Aleae, Cardano frequently makes use of an erroneous principle, which Ore called a “reasoning on the mean” (ROTM) (Ore 1953, p. 150; Williams 2005), to deal with various probability problems. According to the ROTM, if an event has a probability p in one trial of an experiment, then in n trials the event will occur np times on average, which is then wrongly taken to represent the probability that the event will occur in n trials. In our case, we have p = 1/6 so that, with n = 3 throws, the event “at least a six” is wrongly taken to occur an average np = 3(1/6) = 1/2 of the time. But if X is the number of sixes in three throws, then X ˜ B(3,1/6), the probability of one six in three throws is 0.347, and the probability of at least one six is 0.421. On the other hand, the expected value of X is 0.5. Thus, although the expected number of sixes in three throws is 1/2, neither the probability of one six or at least one six is 1/2.

We now move to about a century later when the Chevalier de Méré⁴ (1607–1684) used the Old Gambler’s Rule, leading to fallacious results. As we shall see, the Old Gambler’s Rule is an offshoot of ROTM. The Chevalier de Méré had been winning consistently by betting even money that a six would come up at least once in four rolls with a single die. However, he had now been losing on a new bet, when in 1654 he met his friend, the amateur mathematician Pierre de Carcavi (1600–1684). De Méré had thought that the odds were favorable on betting that he could throw at least one sonnez (i.e., double six) with 24 throws of a pair of dice. However, his own experiences indicated that 25 throws were required.⁵ Unable to resolve the issue, the two men consulted their mutual friend, the great mathematician, physicist, and philosopher Blaise Pascal (1623–1662).⁶ Pascal himself had previously been interested in the games of chance (Groothuis 2003, p. 10). Pascal must have been intrigued by this problem and, through the intermediary of Carcavi,⁷ contacted the eminent mathematician, Pierre de Fermat (1601–1665),⁸ who was a lawyer in Toulouse. In a letter Pascal addressed to Fermat, dated July 29, 1654, Pascal says (Smith 1929, p. 552),

He [De Méré] tells me that he has found an error in the numbers for this reason:

If one undertakes to throw a six with a die, the advantage of undertaking to do it in 4 is as 671 is to 625.

If one undertakes to throw double sixes with two dice the disadvantage of the undertaking is 24.

But nonetheless, 24 is to 36 (which is the number of faces of two dice) as 4 is to 6 (which is the number of faces of one die).

This is what was his great scandal which made him say haughtily that the theorems were not consistent and that arithmetic was demented. But you can easily see the reason by the principles which you have.

De Méré was thus distressed that his observations were in contradiction with his mathematical calculations. His erroneous mathematical reasoning was based on the erroneous Old Gambler’s Rule (Weaver 1982, p. 47), which uses the concept of the critical value of a game. The critical value C of a game is the smallest number of plays such that the probability the gambler will win at least one play is 1/2 or more. Let us now explain how the Old Gambler’s Rule is derived. Recall Cardano’s “reasoning on the mean” (ROTM): If a gambler has a probability p of winning one play of a game, then in n plays the gambler will win an average of np times, which is then wrongly equated to the probability of winning in n plays. Then, by setting the latter probability to be half, we have

Moreover, given a first game with (p₁, C₁), then a second game which has probability of winning p₂ in each play must have critical value C₂, where

That is, the Old Gambler’s Rule states that the critical values of two games are in inverse proportion as their respective probabilities of winning. Using C₁ = 4, p₁ = 1/6, and p₂ = 1/36, we get C₂ = 24. However, with 24 throws, the probability of at least one double six is 0.491, which is less than 1/2. So C₂ = 24 cannot be a critical value (the correct critical value is shown below to be 25), and the Old Gambler’s Rule cannot be correct. It was thus the belief in the validity of the Old Gambler’s Rule that made de Méré wrongly think that, with 24 throws, he should have had a probability of 1/2 for at least one double six.

Let us see how the erroneous Old Gambler’s Rule should be corrected. By definition, C₁ = [x₁], the smallest integer greater or equal to x₁, such that (− p₂)^x1 = 0.5, that is, x _l = ln(0.5)/ln(1 − p₁). With obvious notation, for the second game, C₂ = [x₂], where x₂ = ln(0.5)/ln(1 − p₂). Thus the true relationship should be

We see that Equations (1) and (2) are quite different from each other. Even if p₁ and p₂ were very small, so that ln(1 − p₁) ≈ −p₁ and ln(1 − p₂) ≈ −p₂, we would get x₂ = x₁p₁/p₂ approximately. This is still different from Equation (1) because the latter uses the integers C₁ and C₂, instead of the real numbers x₁ and x₂.

The Old Gambler’s Rule was later investigated by the French mathematician Abraham de Moivre (1667–1754), who was a close friend to Isaac Newton. Thus, in the Doctrine of Chances (de Moivre 1718, p. 14), Problem V, we read,

To find in how many Trials an Event will Probably Happen or how many Trials will be required to make it indifferent to lay on its Happening or Failing; supposing that a is the number of Chances for its Happening in any one Trial, and b the number of chances for its Failing.

TABLE 1.
Critical values obtained using the Old Gambling Rule, de Moivre’s Gambling Rule, and the exact formula for different values of p, the probability of the event of interest

De Moivre solves (1 − p)^x = 1/2 and obtains x = −ln(2)/ln(1 − p). Fo...