  We consider a game with two piles, in which two players take turn to add $a$
or $b$ chips ($a$, $b$ are not necessarily positive) randomly and independently
to their respective piles. The player who collects $n$ chips first wins the
game. We derive general formulas for $p_n$, the probability of the second
player winning the game by collecting $n$ chips first and show the calculation
for the cases $\{a,b\}$ = $\{-1,1\}$ and $\{-1,2\}$. The latter case was asked
by Wong and Xu \cite{WX}. At the end, we derive the general formula for
$p_{n_1,n_2}$, the probability of the second player winning the game by
collecting $n_2$ chips before the first player collects $n_1$ chips.

Combinatorics