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 book was used as part of a two-semester senior-level class at a private university aimed at undergraduates without a pure mathematics background, with a vector calculus prerequisite. Bain can be considered one of the least rigorous mathematical statistics books available, typically accompanied by an amount of handwaving paralleled only by used car dealership blow-up men. The problem sets are fairly diverse and tend to be cumulative, but in self-study, you will find the given answers lacking as they are often incorrect, and the examples presented in chapters do not always provide the needed tools. 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.