Question is quite straight... I'm not very good in this subject but need to understand at a good level.
For probability theory as probability theory (rather than normed measure theory ala Kolmogorov) I'm quite partial to Jaynes's Probability Theory: The Logic of Science. It's fantastic at building intuition behind the rules and operations. That said, this has the downside of creating fanatics who think they know all there is to know about probability theory.
A First Course in Probability by Sheldon Ross is good.
If anybody asks for a recommendation for an introductory probability book, then my suggestion would be the book by Henk Tijms, Understanding Probability, second edition, Cambridge University Press, 2007. This book first explains the basic ideas and concepts of probability through the use of motivating real-world examples before presenting the theory in a very clear way. I found a nice feature of the book the fact that simulation is deliberately used to develop probabilistic intuition. The book also discusses more advanced topics you will not easily find in other introductory probability books. The more advanced topics include Kelly betting, random walks, and Brownian motion, Benford's law, and absorbing Markov chains for success runs. Another asset of the book is a great introduction to Bayesian inference.
I wouldn't know which book is the best, because I've only used two when I was taking a course in probability, but if you'd prefer videos , I'd suggest:
MIT 6.041SC Probabilistic Systems Analysis and Applied Probability course , which is available in MIT OpenCoursWare for free :
Here is a list of great books to own to learn probability & statistics. Some on the list like programming in R are great add-on stuff to know.
I happened to take an introductory course on probability and statistics on two different universities. In one they used a horrible book, and in the other they used a truly amazing one. It's rare that a book really stands out as fantastic, but it did.
Probability and Statistics for Engineers and Scientists by Ronald E. Walpole, Raymond Myers, Sharon L. Myers and Keying E. Ye.
The version number doesn't matter, just find an old version second hand.
It is really thorough, takes one definition at a time, and builds on top of that. The structuring and writing is top class, and the examples are well chosen.
Don't worry if you are not an engineer. When using examples they have taken them from the domain of engineering, eg "A factory produces so and so many items per hour, and only so and so many can be broken, ...", instead of using examples from fx social science or economics. But they don't involve engineering science such as statics, aerodynamics, electronics, thermodynamics or any such things. This means that everyone can understand the book, it does not even help to have an engineering background. Perhaps the examples are more appealing/interesting to engineers, but that's all.
While not a book, Sal Khan's site: http://www.khanacademy.org/ offers dozens of short videos that provide introductions to probability and statistics. Many of the videos even have problem sets associated with them. Khan provides accessible and often intuitive explanations.
He also has extensive video lessons on algebra, linear algebra, calculus, and geometry as well as physics.
All for free.
Find a discussion on this forum which explores pro's and con's about Khan at:
The best book that I have ever read for undergrad and grad students is Intuitive Probability and Random Processes Using MATLAB.
"Lectures on probability theory and mathematical statistics" by M. Taboga. While pretty elementary, it provides proofs of all the main results in probability theory, something you would not find in most other elementary textbooks. It also has plenty of solved exercises and examples. There is a free digital version of this book available at https://www.statlect.com
I found that nobody has pointed out Joe Blitzstein's fantastic book and lectures. the lectures are intuitive and professor Blitzstein does a really good job of explaining the causes behind certain phenomenons in probability, not just mentioning the phenomenon itself, like why do we care about specific distributions, what is the interpretation of a number between 0 and 1 as a probability. he also provides context to each problem for example he mentions how the problem arose? was it a historical problem about gambling or was it a computer science problem or an information theoric one or...?
you can find both the book and lectures in here.
A bit surprised that neither of these have been mentioned. Grinstead & Snell's Introduction to Probability is a good comprehensive introduction, and after that Feller's Introduction to Probability Theory and its applications is a very good serious treatment after introductory probability.