Chances Are by Michael Kaplan Read Free Book Online

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 n th row of the triangle, 2 n , 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.

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    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.
    â€”Abraham de Moivre, Doctrine of Chances
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    N ewton, 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