SLLN tosses of a balanced die

Question

Consider a suquence of tosses of a balanced die. Let $X_k$ be the result of the $k-$th toss and $Y_n=\max_{k\le n} X_k$.

I would like to determine almost surely convergence

1. Find the convergence almost surely of $(Y_n)_{n\ge1}$.

If $Y_1=\max\{X_1\}$, $Y_2=\max\{X_1,X_2\},\ldots,Y_n=\max\{X_1,\ldots, X_n\}$

since $ X_1,\ldots,X_k $ are independent random variables with the same distribution, then for some $j\in\{1,\ldots,n\}$, $Y_n=\max\{X_1,\ldots,X_n\}=X_j$, then the expected value $$E(Y_n)=E(X_j)$$ but all the random variables have the same distribution, hence the same expected value $(E(X_j)=\frac{7}{2})$.

With that, we can use the SLLN but I'm not sure if that's correct.

Answer 1

$E(Y_n)$ has nothing to do with this. Clearly if $X_k=6$ then $Y_n=6\ \forall n\geq k$ and then $Y_n\to6$. Therefore, we just need to show $\Pr(X_n\neq6,\ n\geq1)=0$. I leave that to you.