The concept of picking just one point at random from a probability distribution is not a rigorous mathematical concept, although maybe it could be made rigorous. Or maybe for some reason it cannot be. One bit of mathematical folklore is that there's no such thing as a random integer, if all integers are treated symmetrically. I have long felt skeptical of this claim. Here is one reason: Let ๐ผ denote an irrational number. For definiteness, say ๐ผ = sqrt(2). Let C = R/Z denote the (unit-circumference) circle group. Let g_n denote the element g_n = n*๐ผ (mod 1) in C. Then G = {g_n in C | n in Z} forms a countable dense subgroup of C. There is a well-defined group isomorphism J : G โ> Z taking g_n in G to the integer n in Z. The cosets C/G = {x + G | x in C} of G form an uncontable quotient group that is rarely if ever discussed among decent mathematicians. Let A denote a set consisting of exactly one element from each coset x + G. Then the subsets A_n = A + g_n of G form a partition of G by countably many sets that are rotated versions of each other. The mapping phi : C โ> Z defined as phi(x) = n where n is the unique integer such that x belongs to A_n. Finally, since it is generally accepted that one can select a point at random from the uniform distribution on the unit interval [0, 1), which we identify with C, suppose this is done and the point selected is x. Then a random integer is phi(x). By symmetry, all integers are treated equally. โDan