roll a die an infinity of times and probability that the maximum among all the numbers obtained is 5

Question

I'm a little beat confused about the next problem:

Suppose we roll a die an infinity of times. Calculate (and justify) the probability that the maximum among all the numbers obtained is 5. Here we assume the coin is fair.

I'm confused; I was thinking in defining the event $A_{n}$ as the event that from $1$ to $n$ dice rolled we have as maximum the number $5.$ So we have $A_{n}\subset A_{n+1}$ for all $n\in\mathbb{N}.$ Therefore $P(\bigcup_{n\in\mathbb{N}}A_{n})=\lim_{n\rightarrow\infty}P(A_{n})=0,$ but I'm not sure if my argument is good or wrong.

Any kind of help is thanked in advanced.

Answer 1

If $A_n$ is the event that the maximum in the first $n$ die-rolls is $5$, then it is not true that $A_n \subset A_{n+1}$. For example, if the maximum in the first four rolls is $5$ but the fifth roll is a $6$, your outcome is in $A_4$ but not in $A_5$.

Suggestion: Consider instead the event $B_n$ that there are no $6$'s in the first $n$ rolls. This has $B_{n+1} \subset B_n$, and $A_n \subset B_n$.

Answer 2

The probability that the maximum is 5 is $\le$ the probability that a 6 never comes up. This is 0 probability for an infinite number of rolls.