Hitting k primes by dice rolls

2025/02/12 by Noga Alon, Alon, Noga, Yaakov Malinovsky +5
Mathematics · Physics and Astronomy · #History and Theory of Mathematics #Advanced Mathematical Theories and Applications #Benford’s Law and Fraud Detection

paper · pdf · doi:10.48550/arxiv.2502.08096

Abstract

Let S=(d₁,d₂,d₃, … ) be an infinite sequence of rolls of independent fair dice. For an integer k ≥ 1, let L_k=L_k(S) be the smallest i so that there are k integers j ≤ i for which ∑_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 L₁ is close to 2.43. Here we show that for large k, the expected value of L_k is (1+o(1)) klog_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 ≤ 100.

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