Infinite machines, infinite values, infinite time, same number?

There is a chance. It's a 0% chance, but it's still there (I know that seems weird, but we're dealing with infinite options here, so it's going to be weird). There's no telling when it will happen, because any upper bound on the time and the chance drops to 0% (this time 0-in-whatever for real, as in no chance at all). EDIT: Note that this is the sort of thing many string theorists propose. Right now, given what we actually "know" about string theory and given the mathematics behind string theory, there are about 10^500 possible versions of string theory available to test and no place to begin each one (to put it in perspective, there are "only" about 10^80 particles of matter in the universe, so testing string theory as it currently is would be beyond impossible for us). Basically, this means that string theory is not much of a science and it irks other physicists off when they lose out on research funding to a string theorist's idea because their research isn't considered as cool as string theory research.

Actually, the answer will depend on "how many" machines we have. Suppose we can label each machine with a unique natural number (1, 2, 3, and so on...), then the probability of them choosing the same real number in the first second is 0. Also, the probability of them ever choosing the same real number at any given second is 0. Suppose we can label each machine with a subset of the real numbers, so that for every subset, there's some machine with that label, then the probability of them choosing the same real number in the first second is 1. Therefore the chance of it occurring will be somewhere between almost surely not happening, to almost surely happening, depending on "how many" machines we have. It's also interesting to note letting the process go on forever, checking it each second, has no effect whatsoever on the likelihood. --------- By the way, if you know about cardinality, the above is easy to see, but you might wonder why I ignored the cardinality of the continuum. Well, in short, it's because I don't know the answer to that question... I'm not even sure if there is an answer.

Actually, the answer will depend on "how many" machines we have. Suppose we can label each machine with a unique natural number (1, 2, 3, and so on...), then the probability of them choosing the same real number in the first second is 0. Also, the probability of them ever choosing the same real number at any given second is 0. Suppose we can label each machine with a subset of the real numbers, so that for every subset, there's some machine with that label, then the probability of them choosing the same real number in the first second is 1. Therefore the chance of it occurring will be somewhere between almost surely not happening, to almost surely happening, depending on "how many" machines we have. It's also interesting to note letting the process go on forever, checking it each second, has no effect whatsoever on the likelihood. --------- By the way, if you know about cardinality, the above is easy to see, but you might wonder why I ignored the cardinality of the continuum. Well, in short, it's because I don't know the answer to that question... I'm not even sure if there is an answer.