Can you be more specific about your needs, particularly about the level?
As you say that are many different books.
As a general text I prefer, Probability by Jim Pitman. It is just a good all around college level text.
If you want more statistics then try: Mathematical Statistics by Larsen and Marx. This has good section on counting.
A higher level text is Probability: Theory and Examples by Rick Durrett. It is excellent on: Random Walks, Martingales, Markov Chains, Ergodic Theorems.
The text by D.L. Minh, Applied Probability Models is packed with high level applications and examples.
You may be looking of a graduate level text?
My College uses the book, Probability: A First Course.
I ordered it, but never read it because I never liked probability. I just have it because I am taking Mathematical Probability next semester in hopes of actually liking it.
Note, though it is a "first course" it is a first course for mathematically serious students. So it is an adnvanced book.
And it has bad reviews because most people who ordered it thought it would be an easy book (non-mathemations learn probability too) based on its name.
But that probability class that you were talking about is probably a graduate course.
nominaly an undergraduate course (though I know at least one person from
another university who was given a master course grant to do Part 3 as a
one year course after getting a BSc in Maths)