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Whether or not there are truly random occurences in Nature, we may often treat events as random when these events are made up of many small causes. Brownian motion is the easiest example. A particle undergoing Brownian motion is so jumpy that it has no defined velocity. Einstein and Smoluchowski made a great leap forward when they showed that the position not the velocity of the particle was what could be described. (The position moves "randomly" but total displacement is proportional to the square root of the time.)

In one sense: "Probability Is as Probability Does." (me) Any system satisfying Kolmogorov's axioms may be treated probabilisitically. (There are other axiom sets: de Finetti, Khinchine, von Mises [brother of Ludwig], etc.; these also work but the questions are subtly different.) Jon von Plato has a (rather technical) book on the subject.

Several people have looked at "randomly" occuring events to establish how one should deal with things. Nicole d'Oresme (1323-1382) (No, it wasn't a girl's name then.) was interested in the permanence of the solar system and was opposed to the Stationary Earth Theory as accepted at the time. (He later changed his mind.) Oresme showed that if the rotation period of the earth (day) and the revolution about the sun (year) were not comensurable, then eclipses would happen at different spots on the earth. He also pointed out that the sun would never be in the same place over the earth at an equinox. In addition he was of the opinion that non-comensurablilty was more likely than comensurablity. (Comensurable number have a rational ratio.)

Continuing toward modern times, it was shown by several (Kroneker for example) that on a square billards table, if the tangent of the x and y velocity components of a particle were rational, the particle would trace out a periodic pattern. (Reflecting boundary.) If irrational, the paths would be dense. Dense paths allow time averages to be equal to space averages. (von Neumann, Birkoff, etal.) This means that the time spent in a section of the table would be proportional to the area of that section. If one makes a square table with two opposite side replaced by semicircles, even rational velocities lead to dense paths. (Except for those exactly parallel or vertical to the sides.)