I think the analogy includes and infinite amount of time or an infinite number of randomly typing monkeys (or random letter generators is more correct as you note).
In one case you would get this result an infinite number of times over eons In the second case you would get it an infinite number of times in a few minutes.
Infinity isn't just a big number.
I am pretty darned sure you are wrong when you allow for infinite time. Even 10
50 is a miniscule, vanishingly small amount compared to infinity time..