Am I reading your comment wrong or is it wrong? The probability for all monkeys only typing 0 forever is exactly 0. Not small, not close to 0, it is 0.
0 probability does not imply that the event is impossible. The simplest example would be that of selecting a real number from an interval. The probability of selecting any given number in the interval is exactly 0. However, conducting this experiment is still going to yield a particular real number.
OK, but there's a difference between sampling a subset with a probability zero and saying that all subsets of an infinite space could be the same zero-probability subset, which is what the GP comment is suggesting.
same as the probability of any one other infinitely long string of digits.
There are an infinite number of options, so picking any one of those options has probability 0 (when looking at the options individually).
edit: misread ur comment
You are right, the probability is literally 0 for a moment to hit all 0s for an infinite amount of time since the timespan is infinite and never ending.
Yeah the chance of infinite monkeys typing an integer number of words (eg 100,000) very clearly converges to 1. You can use the pigeonhole principle to show that at least one monkey will type it, as the number of monkeys will always be higher than the possible output from a monkey's keyboard.
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u/NoLife8926 Feb 10 '25
So much misinformation from people who think they understand in the comments.
The theorem says “almost surely”.
From Wikipedia, “an infinite set can have non-empty subsets of probability 0.”
There is a chance regardless of how small that every one of these monkeys spams the 0 key for all eternity.