Probability of sum of dice is composite Let n dice be rolled. Let $S_{i}$ be the sum of the first $i$ rolls for $i=1...n$
Find $Prob($All $S_{i}$ are composite) as $n$ tends to ∞
My guess is 0 but how can I prove this? Or if I'm wrong how do I proceed?

Consider the first $n$ rolls for sufficiently large $n$. Let $$\mathbb{P} = \{0\}\cup\{ 1\leq i \leq n-1\mid \text{there is at least one prime in the range }[S_{i} + 1, S_{i} + 6]\}$$
Since there are more than $\frac{n}{\ln n}$ primes in the range $[1, n]$ for $n \geq 17$ (see here), the size of $\mathbb{P}$ is more than $\frac{n}{6\ln n}$. Hence, 
\begin{align}
\Pr(S_1, S_2, \cdots, S_n \text{ are all composite})~\leq~\left(\frac{5}{6}\right)^{n / (6\ln n)}
\end{align}
When $n$ goes to $\infty$, the probability goes to $0$.
