Free Chances Are by Michael Kaplan Page A

to her family estate, having left behind his father, an agronomist of clerical origin. The newborn Kolmogorov was swept up from the whistle-stop town of Tambov by his maiden aunt. She created a special school for her talented nephew and his little friends; it had its own magazine in which the young Andrei published his first mathematical discovery at the age of five: that the sum of the first n odd numbers is equal to n 2 , as we too discovered in Chapter 1.
    What immediately struck people who met Kolmogorov was his mental liveliness. He remained interested in everything, from metallurgy to Pushkin, from the papacy to nude skiing. His dacha, an old manor house outside Moscow, re-created the estate school of his youth. Presided over by his old aunt and his old nanny, it had a large library and was always full of guests: students, colleagues, visiting scholars.
    It has been said that it would be simpler to list the areas of mathematics to which Kolmogorov did not make a significant contribution than to describe the vast range of topics he did explore—in depth—in his more than seventy years of productive work. His genius was to connect: he took mathematical ideas, clarified their expression, and then used them to transform new fields. He worked on mathematical logic, linking the classical and intuitionist traditions; he worked on function-space theory, extending it to the mechanics of turbulence; he invented the field of algorithmic complexity, and, with characteristic verve, he hoicked up the tottery edifice of probability and slipped new foundations underneath.
    The basic premise of his system was simple: the probability of an event is the same as the measure of a set . We can use a diagram to make this idea even clearer. Take a rectangle:

    Let’s say that everything that can happen in the system we’re interested in—every possible observation—is represented by a point in this rectangle. The probability measure of the rectangle (which, for flat things like rectangles, is its area) is therefore 1, because we can be certain that any observation is represented within it. If we are interested in, say, the flip of a coin, our diagram will look like this:

    Two possible states, with equal area, having no points in common. The chance of throwing an even number with one die? Three independent, mutually exclusive events, totaling half the area of our rectangle:

    We can see that this model nicely represents a key aspect of probability: that the probability of any of two or more independent events happening is determined by adding the probabilities of each. What about events that are not independent (such as, for instance, the event A , that this explanation is clear—and B , that it is true)? They look like this:

    The probability that this explanation is both clear and true is represented by the area shared between A and B . The probability that it is either clear or true is represented by their combined area—although this is not the same as adding their individual areas, since then you would be counting their shared zone twice. The worrying probability that this explanation is neither clear nor true is represented by the bleak, empty remainder of the rectangle.
    What about conditional probability—such as the probability this explanation is true if it is clear? We simply disregard everything outside the area of A (since we presume the explanation is clear) and compare its area with the area that it shares with B —which, as we know, represents clear and true.
    You may find this all a bit simplistic, especially as someone who has come to it through the complex reasonings of Cardano, Pascal, de Moivre, Laplace, and von Mises. The point, though, is that this basic model can be scaled up to match the complexity of any situation, just as Euclid’s axioms can generate all the forms needed to build Chartres cathedral. We need not think only of two circles; we can imagine hundreds, thousands,