Alternatives to Dmitri Bertsekas' Introduction to Probability and Sheldon Ross' A First Course on Probability
If you want to truly understand the mathematical foundations try first learning a bit of measure theory. A probability space is a special kind of measure space and a random variable is a measurable function. Measure theory is not easy but there are several sources. Folland's book on analysis has a chapter on probability theory and Tao's recent book also related measure theory to probability.
Give Bruce Hajek's notes on Random Processes a try. I found some of the diagrams pretty intuitive and easy to understand even if you are a novice or not. See a snippet from his notes below:
