We offer some summation formulas that appear to have great utility in probability theory. The proofs require recent results from analysis thus far been applied basic hypergeometric functions.

In this paper, we show that Zermelo–Fraenkel set theory with Choice (ZFC) conflicts basic intuitions about randomness. Our background assumptions are the Zermelo–Fraenekel axioms without (ZF) together a fragment of Kolmogorov’s probability theory. Using these minimal assumptions, prove weak form contradicts two common sense probability—both based on simple notions symmetry and independence.