Hitting k Primes By Dice Rolls

Abstract

Let S = (d₁, d₂, d₃, . . .) be an infinite sequence of rolls of independent fair dice. For an integer k ⩾ 1, let Lₖ = Lₖ(S) be the smallest i so that there are k integers j ⩽ i for which ∑ʲ ₜ₌₁ dₜ is a prime. Therefore, Lₖ 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 Lₖ is (1 + o(1))k logₑ k, where the o(1)-term tends to zero as k tends to infinity. We also include some computational results about the distribution of Lₖ for k ⩽ 100.