Question: throw a dice and add the number until the sum reaches or becomes larger than a certain given number \(N\). What is the probability that the number \(N\) be reached, rather than over-shoot?

We need to show that, as \(N\) gets large, the probability converges to a constant number. Then we can calculate this number however we want.

I think the fact that the probability of reaching a number \(N\) converges to a constant that does not depend on the "property" (odd or even, prime or nonprime, etc.) of \(N\) is nontrivial. This could be related to the basic assumption of statistical physics. Ergodicity?