7. Probability and Statistics Soviet Essays - Sheynin, Oscar

[3] I bear in mind the well-known problem about determining the probability that afraction chosen at random will be irreducible. Markov, in his classical treatise, offered itssolution following Kronecker and Chebyshev. It is based on the assumption that all theremainders, occurring when dividing a randomly chosen number N by an arbitrary number a(a < N), are equally probable no matter what is the remainder obtained by dividing N by b, anumber coprime with a. It is not difficult to conclude that the required probability is 6/ 2which is the limit of the infinite product∏ ∞=2 p[1 – (1/p 2 )]where the p’s are prime numbers. However, given the stated assumptions, the probability ofN being a prime is zero which is the limit of the product∏ ∞=2 p[1 – (1/p)].Just the same, the probability of N = 2p, 3p, … is also zero. Therefore, the probability thatN is a product of two primes, equal to the limit of a sum of a finite number of zeros, shouldbe zero. In a similar way, the probability that N consists of three, of four, … factors is alsozero. Therefore, again applying the addition theorem, we find that, in general, the probabilitythat an arbitrary number N is a product of a finite number of prime factors is zero. On theother hand, it is doubtless that a finite number consists of a finite number of factors, so thatwe encounter a contradiction: 0 = 1.Thus, Markov’s assumptions are obviously unacceptable. In essence they are tantamountto supposing that all the values of an integer chosen at random are equally possible. Then,however, the probability of a definite value of that number is 1/ = 0 so that the probabilitythat it will not exceed any number N given beforehand is also zero. However, since a givennumber cannot be infinite, we arrive at the same contradiction: 0 = 1.From the point of view of the theory of probability the result obtained by Chebyshev andMarkov and the conclusion necessarily connected with it that the probability for any numberto be prime is zero, should therefore be considered senseless. And, had we neverthelesswished to insist on its correctness, physicists and statisticians could have rightfully told usthat then our law of large numbers with the Bernoulli theorem that only states that someprobabilities are very close to zero, cannot claim to possess a serious experimental orpractical meaning. Actually, the result obtained should have been formulated as follows: Ifall the values of an integer N lesser than a given number are equally probable, then n can bechosen to be so large that the probability that N is prime will be arbitrarily close to zero. Thisproposition is similar in form to the Bernoulli and the Laplace theorems whereas its shortstatement above would have quite corresponded with such an inadmissible form of theLaplace limit theorem: For any integer number N of independent trialsP(t 0