The probability distribution of the number of distinct outcomes in repeated weighted experiments

Distinct Outcomes in Weighted Experiments

Abstract

The number of distinct outcomes arising in repeated weighted experiments is a fundamental quantity in probability theory, with natural applications in statistics and game theory. In this paper, we pursue two complementary objectives. First, we establish explicit and recursive formulas for the probability distribution governing this quantity, then derive its expected value and variance via two independent methods: a classical combinatorial approach and a probabilistic method based on indicator random variables, joint probabilities, and covariance matrices. We demonstrate the practical relevance of this distribution through applications in weighted bootstrap sampling with minority classes and in an analysis of the NBA Draft Lottery. Second, as a central theoretical contribution, we resolve a conjecture proposed in~\cite{FY} asserting that the expected value of this distribution is maximized when all outcomes are equiprobable, a result consistent with the maximum-entropy principle from information theory. We first support the conjecture through Monte Carlo simulations, which motivate and guide a complete formal proof confirming its validity.