Suppose you roll an $n$ sided die (valued $1,\dots,n$) until the sum is at least $s\in\mathbb{N}$. Which of the integers $s, s+1, \dots, s+n-1$ are you most likely to end up with?



Rolling a die until the sum is at least some number
Denote by $p(k)$ the probability that the cummulative sum is equal to $k$ at some point. If we denote by $f(i)$ the probability that the first sum $\geq s$ is equal to $s+i$ then observe that
$$f(i)=\sum\_{k=s+i-n}^{s-1} \frac{p(k)}{n},$$
in particular this implies that $f(0)\geq f(1)\geq \cdots \geq f(n-1)$.


