What happens to the infinite monkey theorem when there are an infinite number of keys on the typewriter? So what is the probability of a finite string of keys like the works of Shakespeare being typed up. Thanks for any insights.
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You need to pick a probability distribution over the keys. If the number of keys is at most countable and each key occurs with positive probability, then the infinite monkey theorem continues to hold and the proof is the same (any finite string only uses finitely many keys so you can ignore the rest). If one of the keys occurs with probability zero and it also occurs in your string, then the string occurs with probability zero.