Hitting k primes by dice rolls

TL;DR

This work analyzes the hitting time $L_k$ for the event that the partial sums of an infinite sequence of fair die rolls encounter primes at least $k$ times. The authors prove that the expected hitting time satisfies $\mathbb{E}[L_k]=(1+o(1))\,k\log k$ as $k\to\infty$ and establish concentration around this scale via probabilistic bounds, complemented by extensive computational data for small $k$. They provide a Maple-based computational package and data up to $k\le 30$, and propose a more precise conjectured form $\mathbb{E}[L_k]\approx k(\log k+\log\log k+c_1)+c_2$, supported by fits. The paper also discusses extensions to other discrete dice distributions and general target sets, showing the asymptotic behavior is robust to variations, with the constants adapting to the mean and reciprocal of the step distribution.

Abstract

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 $k \leq 100$.

Figures (1)

• Figure 1: Scaled probability density function for the number of rolls of a fair die until visiting the primes $k$ times.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• proof