Intuitively, why does $\lim_{n \to \infty} \frac16 (p_{n - 1} + p_{n - 2} ... + p_{n - 6}) = 2/7$? Blitzstein, Introduction to Probability (2019 2 edn), Chapter 2, Exercise 48, p 94.



*A fair die is rolled repeatedly, and a running total is kept (which is, at each time, the total of all the rolls up until that time). Let $p_n$ be the probability that the running total is ever exactly n (assume the die will always be rolled enough times so that the running total will eventually exceed n, but it may or may not ever equal n).

(c) Give an intuitive explanation for the fact that $p_n \rightarrow  1/3.5 = 2/7 \quad as \quad n \rightarrow \infty$.

From p 17 in the publicly downloadable PDF of curbed solutions.

(c)An intuitive explanation is as follows. The average number thrown by the die is (total of dots)/6, which is 21/6 = 7/2, so that every throw adds on an average of 7/2. We can therefore expect to land on 2 out of every 7 numbers, and the probability of landing on any particular number is 2/7.

That's the line I don't get it, why we can transfer

Here's a way to think of it. Suppose you throw the die 1000 times. What sum will you get? Well, of course, it could be anything from 1000 to 6000, but you expect it to be pretty near 3500, right? So, you expect to get 1000 numbers on the way to the sum 3500. Well, getting 1000 numbers out of 3500 is getting, on average, 2 numbers out of every 7. Landing on 1000 numbers on the way to 3500 is landing on two-sevenths of the numbers. So the probability of landing on any particular number is 2/7.  
