Grew too long for a comment:
Counts are discrete, the fact that they can go to infinity doesn't change that.
What makes a variable discrete:
If you can enumerate the values the variable can take ('call this value the first one, that the second one, ...') and never miss any (associate them one-to-one with the natural numbers), then the variable is discrete.
If the sample space is countable, your variable is discrete.
This means, for example, that even the difference of the square roots of two Poisson variables is discrete, even though for any two distinct values, there's always more values between them.
Variables that are counts ("the number of..."), of course, are countable.
Heck, counts can't take values between the integers; 'on a lattice' is as discrete as they come; but (as mentioned already in the different of the square roots of two Poissons example) you don't have to have a lattice to have a discrete r.v.
(response from comments)
@DilipSarwate I agree it's weird (it's equally as weird for the case I mentioned and for the same reason; I'd have mentioned that example as well, but figured it would be overkill to have two examples). But that's what discreteness really is.
[As for difficulty finding probabilities from the pmf, consider $X$, $Y$ and $Z$ all geometric. $X + \pi Y + \sqrt 2 Z$ is plainly discrete, yet some probabilities that may arise for it could be awkward to evaluate also. That doesn't suggest it's ill-defined, only that it's not nice to work with.]
There are an infinite number of ways of assigning probability to each of the rational values in [0,1] of course. Here's one example:
- take $X$ as Poisson $\mu$, $Y$ as (independent of $X$) Poisson $\lambda$, where we condition on $X+Y\neq 0$ and then $V = X/(X+Y)$ is a random variable on the rationals in $[0,1]$. The probability you ask for is just $P(X\leq Y)$ and could be computed in that instance [NB Aside from excluding $X+Y=0$, $X-Y$ has a Skellam distribution]; variables like this really can arise in practice, and even when you can't do the probability calculation algebraically (though in this case you could at least do it numerically to effectively any required bound on accuracy), you can always simulate the required calculations from the definition; the things are well-defined even in cases where they're hard to compute directly.
[Of course, we could order the rationals using the usual diagonalization type argument - only one of an infinite number of ways of doing so - and just assign probabilities from any distribution on the natural numbers (don't ask me what practical use that could have, I don't think it has one) ... while that might make evaluating the probability that you ask for difficult to evaluate (at best), that doesn't mean it's necessarily ill-defined. Even so, since this doesn't seem to have practical use, I am not sure it's necessarily a big concern that it's hard to compute.]
@DilipSarwate I like to make things as simple as possible without saying something that's false. The problem is, I don't think that it is actually true that it is always possible to find an interval $(a,b)$ such that $x∈(a,b)$ and $X$ does not take on any other value in $(a,b)$. In fact, I believe the two examples discussed here are counterexamples - I think for both there's no interval where $a<b$ that won't have probability.
That is, there are discrete cases where there's no empty finite interval - as soon as it has any size at all, there's some points in there.
Edit: the Wikipedia article on Discrete random variables seems to agree -
In cases more frequently considered, this set of possible values is a topologically discrete set in the sense that all its points are isolated points. But there are discrete random variables for which this countable set is dense on the real line (for example, a distribution over rational numbers).
That is, while usually we deal with cases where points are isolated, it's not always the case.
I made the admittedly complicated comment in the later part of my answer because one of the earlier comments looks to me like it was saying something untrue (in fact the same thing that you now suggest).
I'd make a simpler statement in response if I could see a good one that I wasn't pretty sure was wrong.
Is there a simple way to say it?