De Moivre saw that his curve was not only the expression of the laws of probability, but the sign by which previously incoherent data revealed their subjugation to those laws.
As it is thus demonstrable that there are in the constitution of things certain laws according to which events happen, it is no less evident from observation that those laws serve to wise and benevolent purposes. . . . And hence, if we blind not ourselves with metaphysical dust we shall be led by a short and obvious way, to the acknowledgement of the great MAKER and GOVERNOUR of all, Himself all-wise, all-powerful and good.
De Moivre saw God revealed in the pattern of randomness. Having been banished from the deterministic physical world by Descartes, Divinity returns in the unexpected mathematical perfection of chance events. The bell curve shows the trace of an almighty hand—though, of course, particular cases can lie well off the midpoint. De Moivre himself was just such an outlier. He gained little from his genius; even his curves are now named for Gauss and Poisson. He was always a bit too much or too little for the world he lived in, but his own series at last converged at the age of eighty-seven. According to the story, having noticed that he was sleeping a little longer every day, he predicted the day when he would never awake . . . and on that day he died, as the bill of death put it, of “a somnolence.”
The essence of calculus is its ability to conjure precision out of imprecision. No genie could ever sum all the terms of your infinite series, no guardian angel tell you the exact slope of your curve at a single point—but calculus can give you an answer within any desired degree of exactitude. Nothing real in the eighteenth century, not even astronomy, actually demanded a measurement correct to four decimal places, yet de Moivre could provide answers far more accurate than that, simply by working out approximations with a thick carpenter’s pencil on the back of a discarded ballad-sheet.
The power this technique offered to science obscured, at least for a time, the philosophical problems it brought with it, the deepest of which may have been troubling you since we finished with Pascal’s Triangle. Its beautifully symmetrical rows give us a perfect, Platonic representation of the fall of heads and tails, but we have not been talking about anything that actually happens in this world. Our constructs, beautiful as they may be, are descriptions of the behavior of chance in the abstract, as true (or, rather, as valid) at any point in space or time as here and now. To achieve this validity, they depend on the unique ability of mathematics to map the unobtainable and inaccessible: sums of series that go on forever, distances that collapse to zero.
Thus, we get our results by pointing—albeit precisely—at places we can never reach. What has more recently been called “the unreasonable effectiveness of mathematics in the natural sciences” is bought at the price of making a mental leap into the abstract at the point of recording an observation: we assign it to a model universe, at once simpler and purer than the one we inhabit. Dice, after all, are not inherently random: they are a very complicated but determined physical system. Coins need not be ruled by chance; a competent amateur magician learns to flip only heads, pinning us forever to one side of the normal distribution. So when we take up the tools of probability and use them on experience, what are they actually telling us? Something about the real world and its hidden structure? Something about the nature of observation? Or something about our own judgment? These questions are not rhetorical; and if they worry you, you are in the best of company.
Such worries, however, were far from anyone’s mind at the height of the Enlightenment, when the mutually supporting advances of mathematics and the natural sciences convinced most cultured observers that calculus was not just the grammar but the content of Reason. Mathematics was the natural model for the self-evident truths which were such a distinguishing feature of Enlightenment belief, from Newtonian physics to revolutionary politics.
The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance.
The speaker was Pierre Simon, Marquis de Laplace, at once the most brilliant and the most fortunate scientist of his generation. Son of a Norman smallholder, from his arrival in Paris in 1767 at the age of eighteen until his death in 1827—through all the years of discord, revolution, slaughter, empire, war, restoration, and again discord—Laplace never knew misfortune or danger. Ennobled by Napoleon, promoted to marquis by Louis XVIII, he proceeded from honor to honor in an imperturbable orbit.
Laplace’s abilities straddled the crystalline purity of first principles and the messy particularity of actual phenomena. Beyond his virtual invention of the calculus of probability, he made significant mathematical contributions to the sciences of gravitation, thermodynamics, electricity, and magnetism. His accomplishments in physical astronomy were legendary; in effect, the science had to wait for new data before it could have anything further to say.
The fullest expression of Laplace’s contribution to the theory of probability is his Théorie analytique of 1812. Although he said that probability was simply common sense expressed in mathematical language, his theory required the creation of an entirely new branch of calculus, and is presented so densely that it still daunts professional mathematicians. His assistant remembered the great man himself wrestling for an hour to retrace the mental steps he had dismissed with just one “il est aisé à voir.” Easy to see, he meant, in retrospect—blossoming from first principles by agreed forms of reasoning.
Laplace’s universe is entirely determined. He famously claimed that if it were possible for a single intelligence just to know the current position and velocity of all particles, every future event would be predictable—it is only we, with our puny senses, our inefficient instruments and our hasty judgments, who fail to anticipate its revolutions.
Probability therefore is a kind of corrective lens, allowing us, through an understanding of the nature of Chance, to refine our conclusions and approximate, if not achieve, the perfection of Design. The significance of de Moivre’s bell curve is that it describes how what we observe scatters around the pole of immutable law—the “generating function” that represents the ideal behavior of a system. The curve is a tool that cancels out the inescapable error in our data and reveals the one, pure, truth—the secrets not just of the stars but of all physical processes.
Showing the same confidence with which the French Revolution devised the metric system and the decimal calendar, Laplace thought it only logical to extend his principles from the science to morals and politics. Voting, forming assemblies, evaluating legal testimony—all, he felt, would proceed more reasonably if first modeled as the dice-rollings, coin-tossings, lotteries, and urns full of pebbles in probability problems. Probability would damp the terrible fluctuations that can blight the lives of generations, and lead us to the broad sunlit uplands of the Empire of Reason.
Sadly, the fundamental unit of the metric system is not a perfect fraction of the Earth’s circumference, and the French Revolutionary calendar is now remembered only in lobster Thermidor. As for Laplace’s injunction to “examine critically our own opinions and weigh with impartiality their respective probabilities”—how, exactly, should we do this? Well . . . il est aisé à voir.
Nevertheless, by redefining probability from an inherent property of random events to a proportion derived from observations, Laplace helped shape the future of science by prescribing which phenomena it would choose to observe: those that could be entered neatly into probability calculations. And because he put the mind of the observer, rather than the nature of creation, at the center of science, he broke forever Pascal’s hope that faith and scientific inquiry could sit comfortably together.
Not a religious man, Laplace seemed almost to resent the implication that a mere divinity could disturb the perfect regularity of celestial affairs. When Napoleon teasingly remarked that he had seen no mention of the creator of the universe in the Méchanique céleste, Laplace bridled:
“Sire, I had no need of that hypothesis.” He seemed genuinely pained that as great a mind as Pascal’s should give credence to miracles. And so, since probability is a matter of what we can see in this world rather than what could happen anywhere, Laplace made quick use of it to bring Pascal’s Wager fluttering down to earth.
Here is his reasoning—and as always, the crux of the argument is in its initial setting out. Assuming that our choice is between belief in God or not, how did we know we had this choice? Because we had been told about it: witnesses, from evangelists to parish priests, affirmed to us God’s promise that belief leads to an infinity of happy lives. That—not the promise itself—is the observation. Laplace chooses a lottery as his model, which fits well enough with Pascal’s idea that even if the odds against the existence of God approach the infinite, the size of the prize compared with our entry stake makes the game worth playing. Let’s say that Pascal’s lottery is represented by a series of numbered balls, each of which returns that number of happy lives for our stake of one life. We ourselves don’t see the draw, but a witness tells us that we have won the top prize. How credible is that testimony?
Obviously, it’s very much in the interest of these particular witnesses to affirm that we have won, but let us simply assume that the likelihood of their being able to speak with absolute accuracy on such an important matter is, say, one-half. Let us assume moreover the fewest number of happy lives that Pascal claimed would make his wager a guaranteed win: three. There are three balls in the urn; the draw is made. The witness says: “Congratulations! It’s number 3!” Now he is either telling the truth or not. If he is telling the truth, we calculate the probability of observing this by multiplying the innate probability of drawing number 3 (1/3) by the probability that the witness is telling the truth (1/2), and we come up with 1/6.
Now let’s assume the witness is mistaken: some other number came up and your life is wasted. This in itself has a probability of 2/3, which we multiply by the chance (1/2) that the witness is untruthful: our likelihood of discovering this as we trudge the path to Purgatory is therefore 2/3 × 1/2 = 1/3.
So we see that number 3 being drawn and hearing the truth about it is only half as likely as hearing that it was number 3 when it wasn’t. We win only one-third of the time—far from Pascal’s absolute certainty.
We are back in the territory of the Blue Cab/Green Cab problem: when something is intrinsically rare or unlikely, it doesn’t matter how truthful or sincere a witness is—the essential unlikelihood prevails. Even if the witness were 90 percent accurate, the chance of being told accurately that you have won 1,000 happy lives is only 9 in 10,000. As the prize for the celestial lottery goes up, the odds of hearing truthfully that you have won it go down—even without considering any multiplying factors for the self-interest of the witnesses.
Laplace’s calculus of probabilities was intended to give us a mechanism by which we could conquer error and see as on shining tablets the essential laws of the universe. Yet nowadays, we see it as applicable only to a degree and only to a few selected problems. How did this powerful paradigm come to grief? In part, it suffered from that which affects so many great and convincing ideas: its very success tempted people to apply it in realms for which it was ill adapted, sciences far removed from its native astronomy: chemistry, biology, the ever hoped-for social sciences. Here, phenomena did not necessarily fall neatly in a normal distribution around a “true” value. New patterns, new distributions appeared that required a new calculus.
One early example of the new curves made necessary by wider observations was named for Laplace’s student, Poisson, a former law clerk who hoped to apply probability to evidence and testimony. Poisson identified a class of events that, like crime, could happen quite easily at any time, but in fact happen rarely. Almost anyone in the world, for instance, could call you on the telephone right now—but it’s highly unlikely that a ring actually coincided with the moment your eye passed that dash. Quite a lot of human affairs turn out to be like this: the chance of being hit by a car in Rome on any one day is very small, although passing a lifetime there makes the likelihood of at least a bump quite high. You might have a real interest in knowing the relative likelihoods of being run into once, twice, or more times.
Poisson’s distribution is most like real life in not supposing that we know the actual probability of an event in advance. We already “know” the probabilities attached to each side of a die; we “know” the laws that should govern our observation of a planet’s path. We can therefore multiply that known probability by the number of trials to give us our probability distribution. But in real life, we may not have that information; all we have is the product of the multiplication: the number of things that actually happened.
Poisson’s curve, more a steeple than a bell, plots this product of probability times number of trials for things that happen rarely but have many opportunities to happen. The classical case of a Poisson distribution, studied and presented by the Russian-Polish statistician Ladislaus Bortkiewicz, is the number of cavalry troopers kicked to death by horses in 14 corps of the German army between 1875 and 1894.
Here are the raw figures:
The total of trials (20 years x 14 corps) is 280; the total of deaths is 196; the figure for deaths per trial, therefore, is 0.7.
The Poisson formula for this would be
where m is the number of deaths per year whose probability we want to gauge. If m = 1, the probability is 0.3476. If this is applied to 280 experiments, the probable number of times one death would occur in any corps during one year is 97.3 (in fact, it was 91). The theoretical distribution is remarkably close to the actual one (which is probably why this is cited as the classical case):
If it happens you are not a cavalryman, what use could you make of this? Perhaps the best characterization of Poisson’s distribution is that, whereas the normal distribution covers anticipated events, Poisson’s covers events that are feared or hoped for (or both, as in the case of telephone calls). Supermarkets use it to predict the likelihood of running out of a given item on a given day; power companies the likelihood of a surge in demand. It also governed the chance that any one part of south London would be hit by a V2 rocket in 1944.
If you live in a large city, you might consider Poisson’s distribution as governing your hope of meeting the love of your life. This suggests some interesting conclusions. Woody Allen pointed out that being bisexual doubles one’s chance of a date on Saturday night; but sadly Poisson’s curve shows very little change in response to even a doubling of innate probability, since that is still very small compared with the vast number of trials. Your chance of fulfillment remains dispiritingly low. Encouragingly, however, the greatest proportion of probability remains packed in the middle of the curve, implying that your best chance comes from seeking out and sustaining friendships with the people you already like most, rather than devoting too much time to the sad, the mad, or the bad alternative. Like staying away from the back ends of horses, this is a way to make the curve work for you.
Poisson’s distribution could be seen as a special case of the standard distribution—but as probability advanced into statistics it came upon many more curves to conquer, if the mathematics and the data were to continue their engagement. Curves spiky, curves discontinuous—scatters that could not be called curves at all, although they were still defined by functions (that is, rules for assigning a single output to any given input). Mathematics spent much of the nineteenth century seeking methods to bind such boojums and snarks, snaring them in infinite series, caging them with compound constructions of tame sine-curves, snipping them into discrete lengths of manageability—getting their measure.
At the turn of the century, the French mathematician Henri Lebesgue brought these many techniques to their philosophical conclusion: a way to assign a value—a measure—to even the most savage of functions. Measure theory, as his creation was called, made it possible to rein in the wilder curves, gauging the probabilitie
s they represented. It offered its power, though, at the price of intuition: a “measure” is just that. It is simply a means whereby one mathematical concept may be expressed in terms of another. It does not pretend to be a tool for understanding life.
By 1900 it was clear that if the counterintuitive need not be false, the intuitive need not be true. The classical approach to probability could no longer conceal its inherent problems. Laplace had founded his universal theory of probability on physical procedures like tossing a coin or rolling a die, because they had one particularly useful property: each outcome could be assumed to have equal probability. We know beforehand that a die will show six 1/6 of the time, and we can use this knowledge to build models of other, less well known, aspects of life. But think about this for a minute: how, actually, do we know these cases are equally probable?
Well, we could say we have no reason to believe they aren’t; or that we must presuppose equal application of physical laws; or that this is an axiom of probability and we do not question it; or that if we didn’t have equally probable cases . . . we’d have to start all over again, wouldn’t we? All are arguments reflecting the comfortable, rational assumptions of Enlightenment science—and all draw the same sardonic, dismissive smile from our prosecutor, Richard von Mises.
Chances Are Page 5