I've just had the stunning realisation that this experiment has been already performed and the answer (to both the Shakespeare and Carlsen questions) has been found to be affirmative. The key point that has been ignored is that after a few million years of miserable failure at the tasks, the monkeys evolve into humans.
What are the chances of winning any game of chess by chance alone?
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@daw55124, aleph-null (the cardinality of the natural numbers) is definitely more than adequate. No need to add unnecessary complications. Also, the number of ordinals is not a number (i.e. cardinal number). If you assume it is you arrive at paradoxes such as the Cantor paradox, so the axioms need to be chosen to avoid this. [Incidentally "the monkees" were a pop group in the 60s. I have no idea whether they are more suited to the task than the monkeys we were previously discussing]