Some infinities are bigger than others, though.
Even if you have countably infinite monkeys typing countably infinite strings for an infinite period of time, there will be an infinite number of strings that the monkeys haven’t typed, that will never be in the set of completed typed strings.
Cantor’s diagonalization proves it.
You don’t need a normal distribution or statistical independence. It just requires that any given key combination remain possible.
No matter how unlikely, anything that is possible will eventually happen in an infinite time.