  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
$T_{b,w}$ denote the number of draws until this happens. Surprisingly, the
expectation of $T_{b,w}$ is infinite for the "fair" initial scenario $b =w=1$,
but finite if $b=2$ and $w=10^9$. In fact, $\mathbb{E}[T_{b,w}]$ is finite if
and only if $b\ge 2$, and the variance of $T_{b,w}$ is finite if and only if $b
\ge 3$, regardless of the number $w$ of white balls. These observations extend
to higher moments.
