Hitting k primes by dice rolls

TL;DR

This paper analyzes the expected number of dice rolls needed until the sum hits a prime number k times, showing it grows approximately as k log k for large k, supported by theoretical and computational results.

Contribution

It provides a new asymptotic estimate for the expected number of rolls needed to reach k prime sums, extending previous understanding for large k.

Findings

Expected value of L_k is approximately k log k for large k

Expected value of L_1 is about 2.43

Computational results for L_k up to k=100

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 $k \leq 100$.