Then there is every second a nonzero chance that this machine (assuming true and not pseudo randomness) will pick, say pi.
No. The probability of picking any particular number from a uniform distribution is 0.
On the contrary, since the works of Shakespeare are a finite string over a finite alphabet (I can formalize this argument if you want), the probability of typing them out after some fixed large number of keystrokes is some nonzero number 𝑝. With 𝑛 monkeys, the probability that at least one will type out the works is 1 − (1 − 𝑝)ⁿ, which goes to 1 as 𝑛 → ∞.
Now, you are right that this does not mean that the works are guaranteed to be typed out. However, it has probability 1, so it’s mathematically “almost certain”.
No. The probability of picking any particular number from a uniform distribution is 0.
On the contrary, since the works of Shakespeare are a finite string over a finite alphabet (I can formalize this argument if you want), the probability of typing them out after some fixed large number of keystrokes is some nonzero number 𝑝. With 𝑛 monkeys, the probability that at least one will type out the works is 1 − (1 − 𝑝)ⁿ, which goes to 1 as 𝑛 → ∞.
Now, you are right that this does not mean that the works are guaranteed to be typed out. However, it has probability 1, so it’s mathematically “almost certain”.