Choosing 1 of N with and without lucky numbers

Abstract

How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We propose a bit-efficient scheme, prove optimality, derive the expected number of coin tosses $e[n]$, characterize its fractal structure, and develop sharp upper and lower bounds, both discrete and continuous. A minor but noteworthy corollary, with real-world examples, is that any lottery or simulation with finite budget of random bits will have a predictable pattern of lucky and unlucky numbers.