Game of pure chance with restricted boundary
Autor: | Ho-Hon Leung, Thotsaporn \\'Aek\\' Thanatipanonda |
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Rok vydání: | 2020 |
Předmět: |
Discrete mathematics
Recurrence relation Applied Mathematics ComputingMilieux_PERSONALCOMPUTING 0211 other engineering and technologies Probabilistic logic Boundary (topology) 021107 urban & regional planning 0102 computer and information sciences 02 engineering and technology Rational function 01 natural sciences 010201 computation theory & mathematics FOS: Mathematics Mathematics - Combinatorics Discrete Mathematics and Combinatorics Combinatorics (math.CO) Negative number Probability-generating function Mathematics |
Zdroj: | Discrete Applied Mathematics. 283:613-625 |
ISSN: | 0166-218X |
DOI: | 10.1016/j.dam.2020.02.016 |
Popis: | 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. Comment: 19 pages |
Databáze: | OpenAIRE |
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