Now if you were instead ahead by 2 games to 0 when the game is interrupted, you could extend this reasoning: “If I had won the next game, I would have gained all 64; if I had lost, we would be at 2 games to 1—which, as I remember, means I should get 48 pistoles. So the fair division is halfway between these possibilities: 56 for me, 8 for him.” Once again, you are being as just as Aristotle; and you are free to go out into the deepening evening, the serene city before you and gold jingling in your purse.
So far, so good. But can we figure out a general law for interrupted games where you have, say, r points still to make and your opponent has s? Yes, but to do so we need to take an excursion . . . to the beach, perhaps.
Curlews ride the buffeting wind—celestial surfers. The clouds spread in regular ripples like a vast satin quilt. The waves curl, spread, and spring back, as if the ocean were shaking out her hair. Simile reveals pattern—one of the deepest human pleasures, a source of excitement and wonder. For Pascal, the mystical prevalence of pattern was evidence of design and spiritual meaning; for scientists, it is an invitation to explore the unknown, promising that the seemingly random has hidden structure.
In art, we play with pattern to make our own significant marks. We can start here, scratching the simplest figure, 1, in the sand. On either side, for symmetry, another 1, thus:
Let’s spread these wings a little wider, using a rule (itself a kind of pattern) for filling the space between them: mark down the sum of the two numbers just above to the left and right:
and so on, filling the sand as we go with our own symmetrical but unexpected design.
What do we have here? It seems, at first glance, to be no more than the sort of doodle found in the margins of all schoolbooks. Look at it more closely, though—as Pascal did in his Traité du triangle arithmétique—and you will see wonders of pattern. Let’s start by skewing our triangle slightly, to make its rows, columns, and diagonals a little more clear.
Pattern reveals itself first as a lesson in counting. The left column counts as infants do: “This one and then this one and then this one”; the second as grownups do, adding up as we go. The third column lists what are called triangular numbers, that is, the numbers of dots needed to construct equilateral triangles like these:
(or the number of figures, row by row, in Pascal’s own triangle: 1, 3, 6, 10 . . . see?). The next column lists pyramidal numbers, the number of equal-size stones needed to build a regular pyramid with a triangular base—and so on through Fibonacci series, fractal patterns, and further delights of complexity.
The second trick of the triangle may have been discovered by the ancient Hindus and Chinese, and was certainly known to Omar Khayyám (an excellent mathematician as well as poet, tentmaker, and philosophical toper). You will probably remember from school how easy it is, when squaring a binomial (a + b), to forget to include all the relevant combinations in your multiplication: (2 + 3)2 does not equal 22 + 32 and so (a + b)2 cannot equal a2 + b2; instead, it equals:
To calculate (a + b)3, and so on, you would need to include yet more combinations of terms:
Does anything about the numbers in bold—the binomial coefficients—look familiar? Yes, indeed. If you want to know the coefficient of any term in the expanded form of (a + b)n, simply count down n rows in Pascal’s triangle (taking that first 1 you drew as row 0) and count across.
This is a pleasant discovery, but why does it work? Because, in multiplying out your binomial, you have to multiply each term by all the others and then sort the results into groups. In the expansion of (a + b)5, how many groups with 5 a’s are there? One; but there are five groups with 4 a’s and 1 b and ten with 3 a’s and 2 b’s. So the coefficients tell you how many combinations of a and b you can make when you take them in groups of 1, 2, 3, or more.
We are now nearly ready to return to the gaming table and claim our share of the winnings. Play has stopped at a point where you are r points short of victory and your opponent s points short. If you were to go on with neither player gaining an advantage, the longest that play could continue would be another r + s—1 plays, since those are all the points available before one of you must win. Let’s call this largest number of plays n. Now, on each one of these n attempts, you could win or your opponent could win; so the universe of all possible plays can be represented as (1 + 1)n: your potential point and your opponent’s potential point through your n remaining potential games. Given that your opponent had s points to make when the game is interrupted, how much of that universe of 2n potential points is rightfully yours?
(1 + 1)n has a familiar form: it’s a binomial. So if we count n rows down Pascal’s triangle (again, taking the top row as zero), we should find its expansion. And since a and b in this case both equal 1, the only significant terms in that expansion will be the binomial coefficients.
The first term in the row is 1; this represents the single case in which you win all remaining points and the Venetian wins nothing; the second term represents the n cases in which you win all but one of the remaining points and your opponent wins one. Add this term to your total and go on, counting across and adding the coefficients until you get to the sth term of the expansion. From here on to the end, the territory belongs to your opponent: the coefficients represent the number of different ways he can win starting with s points to go. This is your point of division; by comparing the sum of the total number of ways you could have won with the total universe of points, 2n, you get the proportion for dividing the stakes.
Returning to the Venetian, still waiting impatiently: you were within 1 point of winning, but he needed 3. Therefore, s = 3. The total number of points that could be played (n) is 1 + 3 - 1 = 3. The universe of points is therefore 23 = 8. If you count down to the third row of the triangle, you find four coefficients: 1, 3, 3, 1. Since s = 3, you can add the first three coefficients to your total: 1 + 3 + 3 = 7. Compare this with the total universe of points, 8, and you find you have the right to 7/8 of the stakes, or 56 of the 64 pistoles—just as you found before.
In 1654, the same year that God approached him in the form of fire, Pascal listed his accomplishments in a memorial to the Academy in Paris:
... the uncertainty of fortune is so restrained by the equity of reason, that each of two players can always be assigned exactly what is rightly due. . . . By thus uniting the demonstrations of mathematics to the uncertainty of chance . . . it can take its name from both sides, and rightly claim the astonishing title: the geometry of chance.
The geometry of chance: the same geometry Descartes had made interchangeable with equations. The spell of Pascal’s triangle is not just in its elegant array of numerals: each line, if plotted as values on a graph, describes a shape; and successive lines describe that shape with greater and greater precision.
It is this shape that now governs our lives, that defines our normality: the normal, or standard, distribution—the bell curve. The bell curve? How does this come from dicing with Venetians? Because the game we were playing was much more important than it seemed. Winning and losing is not simply a pastime; it is the model science uses to explore the universe. Flipping a coin or rolling a die is really asking a question: success or failure can be defined as getting a yes or no. So the distribution of probabilities in a game of chance is the same as that in any repeated test—even though the result of any one test is unpredictable. The sum of all the numbers on the nth row of the triangle, 2n, is also the total of possible answers to a yes-or-no question asked n times. The binomial coefficients, read across the row, count the number of ways either answer can appear, from n yeses to n nos. If, as here, a perfect bell curve arises from your repeated questioning, you will know that the matter, like Pascal’s game, involves a 50-50 chance.
A game, though, must have rules. How can we try our skill or strength against each other if every trial is different? This is the secret weakness of the method Pascal revealed: we must show we were always playing the same game for the scores to count. That’s a straightforward
task as long as we stay with dice and coins—but as questions become deeper, it grows ever harder to prove that test n is truly identical to test 1. Think, for instance, of asking the same person n times the most significant yes-or-no question of all: “Do you love me?”
The medieval scholars had a clear path to understanding: every aspect of knowledge came with its own distinct rules of judgment. We, when we want to take advantage of the rigor of science, have to define our problem in a form that’s repeatedly testable, or abandon it. “Why is an apple sweet?”—that’s not a scientific question.
3
Elaborating
The same arguments which explode the Notion of Luck may, on the other
side, be useful in some Cases to establish a due comparison between
Chance and Design. We may imagine Chance and Design to be as it were
in Competition with each other for the production of some sorts of Events,
and may calculate what Probability there is, that those Events should be
rather owing to one than to the other.
—Abraham de Moivre, Doctrine of Chances
Newton, apparently, would fob off students who pestered him with mathematical questions by saying, “Go to Mr. de Moivre; he understands these things better than I do.” Abraham de Moivre was a Huguenot refugee who had arrived in London in 1688 with no other patrimony than a Sorbonne education in mechanics, perspective, and spherical trigonometry. It was both more than enough and not nearly enough; for 66 years he didn’t quite make ends meet—publishing this and that, tutoring the sons of earls, helping insurance agents calculate mortality, and selling advice to gamblers on the odds.
De Moivre’s new technique for calculating odds was algebra: consider how powerfully it deals with de Méré’s first problem—how many throws of two dice you need to have an even chance of throwing a double-six. This being algebra, he does not start with one trial and then scale up, he boldly puts an x where we expect to find our answer and takes the most general form of the problem: if something happens a times or does not happen b times during x trials, then we can say, putting the power law into general terms, that the chance of its not happening in every trial is:
We want to find the number of trials where the chance that a double-six will not happen is even, or 1/2:
Using the splendid ability of algebra to re-jig equations in simpler forms, we multiply both sides by (a + b)x and by 2, then divide both sides by bx, and get:
For many years, mathematicians would have been stumped at this point. But de Moivre had the benefit of logarithms, which allow us to manipulate equations to isolate exponents like x from the other elements:
Logarithms ease calculation by considering any large number as a base raised to some power (as a million is also 106). Instead of trying to multiply large numbers, we can simply add the powers of 10 that represent them, since 10a X 10b = 10a+b. So if x =10a , then a = log x. De Moivre was using “natural” logarithms, which have as their base not 10, but e, a mysteriously prevalent irrational number whose decimal expansion begins as 2.718281 . . . so 2 is a little less than e, roughly e 0.7.
Once you’ve found the logarithm tables in the back of your old high-school textbook, you would have something good enough for most practical purposes. But de Moivre is not yet done. Instead of a and b, let’s talk about 1 and q, so that we can say the odds against success are q to 1; that makes our basic equation (expressed in odds rather than probabilities):
De Moivre had a general method for dealing with crabbed, knotty expressions like log(1 + 1/q), which involved infinite series: that is, adding together an infinite number of terms, thus:
Er, yes . . . and exactly how is algebra helping us here? The formula looks clear enough, but then so do the instructions “Move this mountain using these tweezers.” De Moivre, however, noticed that, assuming q is big enough, all the terms after the first are so small as to make hardly any difference to the total—particularly since they alternately add and remove progressively tinier and tinier amounts. In fact, they are usually so small that we can dispense with them completely, allowing us to be reasonably correct without being exact. So, having done this major surgery, we are left with:
A quick glance above will remind you that log 2 is roughly 0.7. So, having started with a vague and general question, algebra offers us a startlingly specific result: given big enough odds against something, you can gauge how many trials are needed to have an even chance of seeing it happen by multiplying the odds by 0.7. This will apply to anything: roulette, cards, eclipses of the moon. In the case of de Méré’s dice, the odds against rolling a double-six are 35 to 1 (as you know); 35 x 0.7 = 24.5 trials: exactly the answer Pascal got by calculating cases.
Infinite series and logarithms also gave de Moivre the key to understanding Pascal’s Triangle—and, with it, the bell curve. A few pages ago we blithely said that if you wanted to know, say, the chance that a heads will appear in n tosses of a coin, you need only count n rows down the triangle and a across to find the coefficient. Easy enough for five or six trials, but imagine actually doing it for a series of 1,000 trials. Here are the tweezers—there is the mountain. There are clearly good reasons for asking de Moivre and his algebra to help us.
His attack on the problem took advantage of the two most important aspects of logarithms and infinite series: that logarithms allow you to get the result you need by adding logs rather than multiplying big numbers; and that infinite series can converge. That is (as we saw with the series for 1+ 1/q), although the summing goes on forever, each new term makes progressively less and less difference to the total, so you can see where the series is heading even if you never quite get there. De Moivre then made one of those shifts of viewpoint that define genius in mathematics. Having teased out his formula for the middle term of the nth row of the triangle into an infinite series, he seemed to invite confusion and disaster by expanding each term in that series as an infinite series in itself—wheels within wheels, new worlds riding on fleas’ backs. Then, spreading out this vast quilt of terms, of sums of sums, he suggested that instead of adding across the rows, we should add down the columns, which reveal themselves as converging to neat and orderly formulae. For large numbers of trials, this infinite range of potential calculation collapses into two simple formulae for the approximate values (relative to the total, 2n) of the middle term and any other term t places away from the middle of row n of Pascal’s triangle:
You could, if you wish, use this formula to solve the problem of points; indeed, you would probably have to if you were playing more than, say, a hundred games.
Even if your taste is not for formal mathematics, you have a glimmering here of its entrancing power. This shabby man, ensconced at the window end of a greasy table in Slaughter’s Tavern, had, through the clever manipulation of abstract terms, discovered—or created—a way to describe how things happen the way they “ought”: how Chance scatters itself around the central pillar of Design in the shape of a bell. “If the terms of the binomial are thought of as set upright, equally spaced and at right angles to a straight line, the extremities of the terms follow a curve.” He went on to say: “The curve so described has two inflection points, one on each side of the maximal term.” Let’s look at this curve, basing our example on the numbers from row 14 of Pascal’s Triangle.
It’s still pretty rough, since we have only 15 fixed points, but the form is becoming clear, particularly the “inflection points” that de Moivre mentioned: the places where the convex curve down from the center changes into a concave curve out toward the sides. Those are the most interesting points on the curve; de Moivre calculated their distance from the midpoint of the curve as 1/2√n, for n trials. This number is what statisticians now call the standard deviation, one of their most important terms—and its importance stems from what de Moivre did next.
Since the newly invented calculus gave him the ability to measure the area under the curve, de Moivre decided to see what fraction of that area wa
s taken up by the vertical slice of area between the midpoint and the inflection point. Again, the calculation involved an infinite series that converged to a single, disconcertingly precise figure: 0.341344 (assuming that the total area under the curve equals 1). So the area covered by the curve between its two inflection points is double that: 0.682688.
Once again, the power of formal mathematics has handed us a golden key—but to what? To the orderly expectation of random behavior. It reveals that, for an experiment with equal chances of success or failure, we can expect a bit more than two thirds of our results to be within 1/2√n of our expected middle value. The more trials we make, the bigger n becomes and the closer the total results come to their inherent probability. So, as de Moivre showed, if we toss a coin 3,600 times, the probability is 0.682688 that we will get between 1,770 and 1,830 heads, and the probability is 0.99874—as close to certainty as most mortals know—that we will toss between 1,710 and 1,890 heads. Make yet more trials, and the window will tighten as the curve becomes ever taller and narrower: the behavior of the experiment is inherent in the nature of the curve.
Chances Are Page 4
