> What percentage of students after having passed a probability course can define a random variable?

In a typical probability theory course, all of them, hopefully. I found random variables to be a crucial part of my probability theory class, being used in almost all (if not all) topics after the first month.

There are certainly other approaches to learning probability theory (though perhaps not what you were thinking.) I understand that E.T. Jaynes' book on probability theory, which I have not yet read, does not go the route of random variables.