Two people take turns throwing a fair die, let $M_n$ denote the number of times during the first $n$ throws ($n$ is the total number of throws for both people) that the second person throw beats the preceding throw of the first person.
What is the $\lim_{n \to \infty} \frac{M_n}{n}$ ?
What I tried:
The proportion of times the second throw beats the first person's throw = $\frac{M_n}{n}$
By statistical definition of probability, we get
If total number of occurrences = $n$ ; number of occurrences of a particular event = $f$
Then, Probability of occurrence that event = $\lim_{n \to \infty} \frac{f}{n}$
Therefore, $\lim_{n \to \infty} \frac{M_n}{n}$ = Probability that the second person throw beats the preceding throw of the first person.
Is there any more steps to know more about the limit?