论文标题
边界有限的纯粹机会游戏
Game of Pure Chance with Restricted Boundary
论文作者
论文摘要
我们考虑各种概率游戏，其中包括一两个玩家的堆。在每一轮比赛中，玩家都会随机选择在$ a $ a $ a $ b $不一定是正面的条件下，在他的堆中添加$ a $ a或$ b $ chips。如果玩家发挥作用后有负数的芯片数量，那么他收集的芯片数量将保持$ 0 $，并且游戏将继续。我们考虑的所有游戏都满足了这些规则。当人们第一次收集$ n $芯片时，游戏结束。允许每个玩家以$ s $芯片开始，其中$ s \ geq 0 $。我们考虑了各种$（a，b）$的情况，包括对$（1，-1）$和$（2，-1）$。我们研究了结束游戏所需的回合数量的概率生成功能。我们在$ n $中为此类函数的序列得出了有趣的复发关系，并将这些生成函数写为理性函数。作为一个应用程序，我们得出了游戏的其他统计数据，其中包括结束游戏所需的平均圈数和其他更高的时刻。
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.
