Discussion of Borel's Law
The first is Probability and Life, a 1962 Dover English translation of the French version published in 1943 as Le Probabilites et la Vie. The second is Probability and Certainty, a 1963 Dover English translation of the French version published in 1950 as Probabilite et Certitude. Both of these books are "science for the non-scientist" type books rather than scholarly treatments of the theory of probability.
In Probability and Life, Borel states a "single law of chance" as the principle that "Phenomena with very small probabilities do not occur". At the beginning of Chapter Three of this book, he states:
When we stated the single law of chance, "events whose probability is sufficiently small never occur," we did not conceal the lack of precision of the statement. There are cases where no doubt is possible; such is that of the complete works of Goethe being reproduced by a typist who does not know German and is typing at random. Between this somewhat extreme case and ones in which the probabilities are very small but nevertheless such that the occurrence of the corresponding event is not incredible, there are many intermediate cases. We shall attempt to determine as precisely as possible which values of probability must be regarded as negligible under certain circumstances.
It is evident that the requirements with respect to the degree of certainty imposed on the single law of chance will vary depending on whether we deal with scientific certainty or with the certainty which suffices in a given circumstance of everyday life.
The point being, that Borel's Law is a "rule of thumb" that exists on a sliding scale, depending on the phenomenon in question. It is not a mathematical theorem, nor is there any hard number that draws a line in the statistical sand saying that all events of a given probability and smaller are impossible for all types of events.
Borel continues by giving examples of how to choose such cutoff probabilities. For example, by reasoning from the traffic death rate of 1 per million in Paris (pre-World War II statistics) that an event of probability of 10-6 (one in a million) is negligible on a "human scale". Multiplying this by 10-9 (1 over the population of the world in the 1940s), he obtains 10-15 as an estimate of negligible probabilities on a "terrestrial scale".
To evaluate the chance that physical laws such as Newtonian mechanics or laws related to the propagation of light could be wrong, Borel discusses probabilities that are negligible on a "cosmic scale", Borel asserts that 10-50 represents a negligible event on the cosmic scale as it is well below one over the product of the number of observable stars (109) times the number of observations that humans could make on those stars (1020).
To compute the odds against a container containing a mixture of oxygen and nitrogen spontaneously segregating into pure nitrogen on the top half and pure oxygen on the bottom half, Borel states that for equal volumes of oxygen and nitrogen the odds would be 2-n where n is the number of atoms, which Borel states as being smaller than the negligible probability of 10-(10(-10)), which he assigns as the negligible probability on a "supercosmic" scale. Borel creates this supercosmos by nesting our universe U1 inside successive supercosmoses, each with the same number of elements identical to the preceding cosmos as that cosmos has its own elements, so that U2 would be composed of the same number of U1's as U1 has atoms, and U3 would be composed of the same number of U2's as U2 has U1's, and so forth on up to UN where N=1 million. He then creates a similar nested time scale with the base time of our universe being a billion years (T2 would contain a billion, billion years) on up to TN, N=1 million. Under such conditions of the number of atoms and the amount of time, the probability of separating the nitrogen and oxygen by a random process is still so small as to be negligible.
Ultimately, the point is that the user must design his or her "negligible probability" estimate based on a given set of assumed conditions.
