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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.10246 |
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| _version_ | 1866910381129072640 |
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| author | Fridman, Yehonatan |
| author_facet | Fridman, Yehonatan |
| contents | Consider the following game between a random player R and a deterministic player D. There is a pile of n elements at the beginning. The rules for playing are as follows: In each turn of R, if the pile contains exactly m elements, R removes k elements from the pile, where k is independently identically distributed from {1, . . . , m}. In each turn of D, D removes only one element. The winner is the player that, at the end of its round, has no elements remaining. R starts first to play. This short paper shows that Dn, which is defined as the probability of D winning the game (when is initialized with n elements), approaches 1/e when n increases; and more specifically, Dn = dn/n!, where dn is the n-th derangement number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Random-Player Game and Derangement Numbers Fridman, Yehonatan Combinatorics Probability Consider the following game between a random player R and a deterministic player D. There is a pile of n elements at the beginning. The rules for playing are as follows: In each turn of R, if the pile contains exactly m elements, R removes k elements from the pile, where k is independently identically distributed from {1, . . . , m}. In each turn of D, D removes only one element. The winner is the player that, at the end of its round, has no elements remaining. R starts first to play. This short paper shows that Dn, which is defined as the probability of D winning the game (when is initialized with n elements), approaches 1/e when n increases; and more specifically, Dn = dn/n!, where dn is the n-th derangement number. |
| title | A Random-Player Game and Derangement Numbers |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2402.10246 |