Probability of picking a specific value from a countably infinite set

Solution 1:

I'm assuming that implicit in your question is that you're looking for a uniform distribution. (Otherwise, the statement "picking a specific value from an uncountably infinite set has a probability of zero" is false.)

To answer such questions systematically, you need a clear definition of what you mean by probabilities. You'll find the usual definition e.g. in the Wikipedia articles on probability axioms, probability measure and probability space. The key point there is that probabilities need to be countably additive. This allows you to derive a contradiction from assigning zero probability to elementary events in a countable probability space, but not in the case of an uncountable space. Assigning zero to a singleton set in a countable space leads to the contradiction that the countable sum of the zeros for all the singletons must be $0$ (from countable additivity), but $1$ because it's the probability for the entire space. Note that this has nothing to do with "discreteness" in a topological sense; e.g., it's true for the rationals, independent of whether you regard them as a discrete space or with the usual topology induced by the topology of the reals.