The Second Edition of INTRODUCTION TO PROBABILITY AND MATHEMATICAL STATISTICS focuses on developing the skills to build probability (stochastic) models. Lee J. Bain and Max Engelhardt focus on the mathematical development of the subject, with examples and exercises oriented toward applications.
This can be considered one of the least rigorous and thereby most accessible books on mathematical statistics available. The problem sets are fairly diverse and tend to be cumulative but you will find the given answers occasionally incorrect. As the author was a frequentist statistician, do not expect to find much in the way of Bayesian discussion.
Probably (pun intended) the best pure Probability text out there right now. Provides great insight with sufficient mathematical rigor to validate claims.