# Given an infinite time, can a number be chosen randomly from a set of infinite real numbers?

This is related to the monkeys writing hamlet problem. From my understanding the definition of the probability in this situation is if there is a probability something will happen, given an infinite time it will happen. So with the monkeys, the set the monkeys have to choose from is lets say the alphabet. (I know there is more, but to simplify the question lets make the set size 26). So the monkey has to choose from {a, b,...,z} and continuously chooses at random for infinite time. to get the first letter correct the probability is 1/26. the second is (1/26)^2, and so on. So probability hamlet will be written is (1/26)^n, with n being the letters in the story. This is a probability that is measurable and therefore given an infinite time will happen. Sure, I can understand that. But now the meat of my question:

Given an infinite set of real numbers {-infinity,...,infinity}, a number is chosen at random. A robot is designed to take infinite time trying to guess the number that was chosen, choosing from the set of all real numbers. The robot is truly random at choosing. Choosing any number from the set is 1/infinity, which intuitively to me seems like it is zero. In fact, given a number and epsilon, there is an infinite set {n,...,epsilon}, which theoretically the robot could spend infinite time choosing numbers in that set. However... I was TOLD this was wrong, from 1 professor, and I had another mathematics professor tell me my idea sounds right, but he didn't have the upper division background in probability to give me a good answer. I'm skeptical of the first professors answer, so could someone explain to me whether or not the robot will get the answer?

(My background as of this moment consists of vector calculus and linear algebra).

• There are several problems here. The first is that the robot cannot choose a number uniformly from the whole real number line (though it might be able to non-uniformly). The second is that if the robot chooses a single number in a positive finite time, it cannot choose an uncountably infinite number of numbers in a countably infinite time. – Henry Nov 6 '16 at 21:26

That's too informal. "Given an infinite time" does not make sense. What you can say is: given an experiment with probability of sucess $p>0$, if you repeat $n$ (independent) trials of it and call $P_n$ the probability of having at least one success, then $P_n \to 1$ as $n \to \infty$. You must be careful, still. First, this doesn't say that we have $P_n=1$ after some $n>N$; and you must also remember that $p_E=0$ is not equivalent to "event $E$ is impossible", as $p_E=1$ is not equivalent to "event $E$ must happen".
Anyway, this property is useless for your situation, and your robot won't work (even if it's trying to guess a real number drawn uniformly in the interval $[0,1]$), because the individual probability of success is zero.