Geodesic
Hitting \(k\) primes by dice rolls
Noga Alon ; Yaakov Malinovsky1 ; Lucy Martinez ; Doron Zeilberger
1University of Maryland, Baltimore County
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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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 $k \leq 100$.
DOI : 10.37236/13878
Classification : 60C05, 60G40, 11A41
Mots-clés : independent fair dice, accumulated sum, primes

Noga Alon    ; Yaakov Malinovsky  1   ; Lucy Martinez    ; Doron Zeilberger 

1 University of Maryland, Baltimore County 
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Noga Alon; Yaakov Malinovsky; Lucy  Martinez; Doron Zeilberger. Hitting \(k\) primes by dice rolls. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13878

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