  We consider various probabilistic games with piles for one player or two
players. In each round of the game, a player randomly chooses to add $a$ or $b$
chips to his pile under the condition that $a$ and $b$ are not necessarily
positive. If a player has a negative number of chips after making his play,
then the number of chips he collects will stay at $0$ and the game will
continue. All the games we considered satisfy these rules. The game ends when
one collects $n$ chips for the first time. Each player is allowed to start with
$s$ chips where $s\geq 0$. We consider various cases of $(a,b)$ including the
pairs $(1,-1)$ and $(2,-1)$ in particular. We investigate the probability
generating functions of the number of turns required to end the games. We
derive interesting recurrence relations for the sequences of such functions in
$n$ and write these generating functions as rational functions. As an
application, we derive other statistics for the games which include the average
number of turns required to end the game and other higher moments.
