  We study first passage statistics of the Polya urn model. In this random
process, the urn contains two types of balls. In each step, one ball is drawn
randomly from the urn, and subsequently placed back into the urn together with
an additional ball of the same type. We derive the probability G_n that the two
types of balls are equal in number, for the first time, when there is a total
of 2n balls. This first passage probability decays algebraically, G_n ~ n^{-2},
when n is large. We also derive the probability that a tie ever happens. This
probability is between zero and one, so that a tie may occur in some
realizations but not in others. The likelihood of a tie is appreciable only if
the initial difference in the number balls is of the order of the square-root
of the total number of balls.
