I believe this might be more of a analysis question. (@admin - if you feel the same you may please move it accordingly. Thanks)
We pick a real number randomly from [0,1]. What is the probability it is rational?
I think it is 0. (because of the cardinality stuff) But how to demonstrate it formally?
http://en.wikipedia.org/wiki/Measure_(mathematics)
The idea is that if are disjoint sets, then you can represent the probability of the countable union as the probability of the sum. So when you know the probability of picking a fixed , you can just do this
I hope this is helpful in some way