Hitting a prime in 2.43 dice rolls (on average)

TL;DR

This paper calculates the expected number of fair 6-sided dice rolls until the total sum is prime, providing precise expectation and variance estimates with a simple proof combining dynamic programming, MATLAB, and prime distribution facts.

Contribution

It offers a complete, rigorous solution to a puzzle about prime sums in dice rolls, improving upon an incomplete previous solution.

Findings

Expected rolls until prime sum: 2.43

Variance of the number of rolls is computed

Method combines dynamic programming and prime distribution analysis

Abstract

What is the number of rolls of fair 6-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10^{-4}. This is a solution to a puzzle suggested by DasGupta (2017) in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes.