Free Chances Are by Michael Kaplan Page B

indeed an infinity of measurable subsets of our rectangular sample space, overlapping and interpenetrating like swirls of oil on water. Nor need our space be a two-dimensional rectangle; the same axioms would apply if our chosen measure were the volume of three-dimensional objects or the unvisualizable but mathematically conventional reality of n -dimensional space. And since this idea of probability borrows its structure from set theory, we can do logical, Boolean, calculations with it—well, we can’t, but computers can, since they thrive on exactly those tweezer-and-mountain techniques of relentlessly iterated steps that fill human souls with despair.
    There need be no special cases, cobbled-together rules or jury-rigged curves to cover this or that unusual situation. The point of Kolmogorov’s work is that mathematical probability is not separate from the remainder of mathematics—it is simply an interesting aspect of measure theory with some quaint terminology handed down from its origins in real life.
    Thus embedded, probability—understood as the mathematics of randomness—found again the rigor of deductive logic. True, it appeared at first to be a somewhat chilly rigor—one that its practitioners were keen to distinguish from the questionable world of applications. William Feller, who wrote the definitive mid-twentieth century textbook on probability, began it by pointing out: “We shall no more attempt to explain ‘the meaning’ of probability than the modern physicist dwells on the ‘real meaning’ of mass and energy.” Joseph Doob, one of the most prominent, if stern, proponents of probability theory, said that it was as useless to debate whether an actual sequence of coin tossing was governed by the laws of probability as “to debate the table manners of children with six arms.” As always, the counselors of perfection demand a retreat from the world.
    That, though, was to reckon without humanity. Our desire to come to some conclusion—even if it’s not certainty—means we are bound to take this specialist tool of probability and risk its edge on unknown materials. Kolmogorov’s legacy is applied every day in science, medicine, systems engineering, decision theory, and computer simulations of the behavior of financial markets. Its power comes from its purity—its conceptual simplicity. We are no longer talking specifically about the behavior of physical objects, observations, frequencies, or opinions—just measures. Purged of unnecessary ritual and worldly considerations, uniting its various sects, mathematical probability appears as the one true faith.
    But having a true faith is not always the end of difficulties. Consider a simple problem posed by the French mathematician Joseph Bertrand: you have a circle with a triangle drawn inside it—a triangle whose sides are equal in length.

    Now you draw a line at random so that it touches the circle in two places: this is called a chord . What is the probability that this chord is longer than a side of the triangle?
    One good path to an answer would be this: line up a corner of our triangle with one end of the chord; we can now see that any chord that falls within the triangle will be longer than a side.

    Since the starting angle of lines that cross the triangle is one-third the angle of all possible chords you could draw from that point, this suggests that the probability that a random chord will be longer than a side of the triangle is 1/3.
    But here is a different approach. Let’s say you take your chord and roll it across the circle like a pencil across the floor.

    We can see that any chord falling within the rectangle built on a side of the triangle will be longer than that side. The height of that rectangle is exactly half the diameter of the circle; so the probability that a random chord will be longer than a side of the triangle is 1/2· Paradox.
    Probability