If I draw a number uniformly between zero and one, what is the probability that they are equal? Mathematically, it should be zero but I don't recall why? Can somebody please help in explaining why the probability should be zero?
For example, there is still a chance, that when generating six random numbers they will be all 1 or all 0? So if the probability is zero then why in digital communications do we assume that the symbols are $iid$ and that if the symbols are 0 and 1, 50% of them will be 1 and the rest will be 0 i.e., why do we assume equi-probable symbol probabilities when theoretically they cannot be equal? In Matlab, I tried my best to generate a large sequence of 0/1 but never did I get equi-probable symbol probabilities.
Am I misunderstanding some thing here?