Hitting k primes by dice rolls
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866929711686352896 |
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| author | Alon, Noga Malinovsky, Yaakov Martinez, Lucy Zeilberger, Doron |
| author_facet | Alon, Noga Malinovsky, Yaakov Martinez, Lucy Zeilberger, Doron |
| contents | Let $S=(d_1,d_2,d_3, \ldots )$ be an infinite sequence of rolls of independent fair dice. For an integer $k \geq 1$, let $L_k=L_k(S)$ be the smallest $i$ so that there are $k$ integers $j \leq i$ for which $\sum_{t=1}^j d_t$ is a prime. Therefore, $L_k$ is the random variable whose value is the number of dice rolls required until the accumulated sum equals a prime $k$ times. It is known that the expected value of $L_1$ is close to $2.43$. Here we show that for large $k$, the expected value of $L_k$ is $(1+o(1)) k\log_e k$, where the $o(1)$-term tends to zero as $k$ tends to infinity. We also include some computational results about the distribution of $L_k$ for $k \leq 100$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_08096 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hitting k primes by dice rolls Alon, Noga Malinovsky, Yaakov Martinez, Lucy Zeilberger, Doron Probability Combinatorics Number Theory 60C05, 11A41, 60G40 Let $S=(d_1,d_2,d_3, \ldots )$ be an infinite sequence of rolls of independent fair dice. For an integer $k \geq 1$, let $L_k=L_k(S)$ be the smallest $i$ so that there are $k$ integers $j \leq i$ for which $\sum_{t=1}^j d_t$ is a prime. Therefore, $L_k$ is the random variable whose value is the number of dice rolls required until the accumulated sum equals a prime $k$ times. It is known that the expected value of $L_1$ is close to $2.43$. Here we show that for large $k$, the expected value of $L_k$ is $(1+o(1)) k\log_e k$, where the $o(1)$-term tends to zero as $k$ tends to infinity. We also include some computational results about the distribution of $L_k$ for $k \leq 100$. |
| title | Hitting k primes by dice rolls |
| topic | Probability Combinatorics Number Theory 60C05, 11A41, 60G40 |
| url | https://arxiv.org/abs/2502.08096 |