reference

This is probably a very nice text book if you already know probability. There are undeniably some insightful examples. However, it is often hard to follow the sequence of topics in the book.

It is at least amusing that the integral is only developed a couple of chapters after expectation has been in use...

This book gives abundant examples and statements that help to deepen your understanding. It does not require much statistic background to follow the book, although sometimes the reasoning is not that obvious to me. But maybe because I am not a math student. I also feel that topics are a bit scattered in the book.

the main problem with Billingsley's book lies in its organization of topics and results.
Yes it has all the standard results that need to be covered in a first (rigorous) course on probability theory and the proofs and exercises are good (thats why the three stars) but it is incredibly hard to study them from this book because of poor organisation which makes for lack of continuity (thats why no more than three).
Stick to Chung (and move to something more specialized thereafter). Unfortunately, Parthasarathy's 1977 Macmillan book is now out of print and only available in libraries ... I find that to be the best book at this level.