2019/11/04 by Norbert Henze, Henze, Norbert, M.P. Holmes +1
Decision Sciences · Mathematics · #00A08 #60E99 #FOS: Mathematics #Game Theory and Applications #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1911.01052
openalex publication_date 2019/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider an urn initially containing b black and w white balls. Select a ball at random and observe its color. If it is black, stop. Otherwise, return the white ball together with another white ball to the urn. Continue selecting at random, each time adding a white ball, until a black ball is selected. Let Tb,w denote the number of draws until this happens. Surprisingly, the expectation of Tb,w is infinite for the "fair" initial scenario b =w=1, but finite if b=2 and w=109. In fact, 𝔼[Tb,w] is finite if and only if b≥ 2, and the variance of Tb,w is finite if and only if b ≥ 3, regardless of the number w of white balls. These observations extend to higher moments.