The thing is that they're not equally likely !
A discrete distribution is not necessarily a uniform distribution (this is what you meant in your bold sentence)
You can define :
And we indeed have , so it's a random variable.
Does this example shed some light on your mistake ?
What does it mean : "choosing one is a continuous random variable" ?Also, the irrational numbers are uncountable... so I suppose that choosing one at random must be a continuous random variable... but this is very counter-intuitive since there is a rational number between any two irrationals...
While working with continuous rv's, the probability of taking a given value, even if it's an integer, or an irrational, is 0.
(because the Lebesgue measure of a singleton is 0... but I guess you didn't learn that yet)