Kolmogorov drew analogies between probability and measure, resulting in five axioms, now usually formulated in six statements, that made probability a respectable part of mathematical analysis. The most basic notion of Kolmogorov’s theory was the “elementary event,” the outcome of a single experiment, like tossing a coin. All elementary events formed a “sample space,” the set of all possible outcomes. For lightning strikes in Massachusetts, for example, the sample space would consist of all the points in the state where lightning could hit. A random event was defined as a “measurable set” in a sample space, and the probability of a random event as the “measure” of this set. For example, the probability that lightning would hit Boston would depend only on the area (“measure”) of this city. Two events occurring simultaneously could be represented by the intersection of their measures; conditional probabilities by dividing measures; and the probability that one of two incompatible events would occur by adding measures (that is, the probability that either Boston or Cambridge would be hit by lightning equals the sum of their areas).
I hadn’t heard of Nautilus Quarterly before. It looks interesting.
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