The strong law of large numbers states that if $X$ is a RV and $X_1, X_2 \ldots$ are independent and identically distributed copies of $X$, and $\overline{X}_n= \frac{1}{n}(X_1 + \ldots + X_n)$, and the first moment of $X$ is finite, then
$P(\lim_{n\to\infty} \overline{X}_n = \Bbb E X) =1$.
But by definition of limit this means that is almost certain that for every $\epsilon$ there exists a $n_0$ such that $n>n_0 \implies |\overline{X}_n - \Bbb E X|<\epsilon$.
What meaning can we give to "the probability that something exists" (in this case $n_0$)?