CURIOSITIES REGARDING WAITING TIMES IN POLYA'S URN MODEL

Abstract

Consider an urn, initially containing b black and w white balls. Select a ball at random and observe its colour. 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.