Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \dots, d\}$. What is the probability that the sequence $s_0, s_1, s_2, \dots$ contains infinitely many primes?
Source: SMMC 2024 A4
Tags: number theory
Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \dots, d\}$. What is the probability that the sequence $s_0, s_1, s_2, \dots$ contains infinitely many primes?