A good book for Probability? I am doing my research in Functional Analysis, especially in "Generalized inverse of Linear Maps".
I have come across Probability by studying only the methods or Distributions(like Binomial, poisson, normal,etc)
Now I wish to study the mathematical background and intuitive way of looking on it.
Can you suggest some probability text book which explains,

$\bullet$ Geometrical ideas of the Probability Distributions. May be using diagrams, graphs, etc.
$\bullet$ Proofs for the distribution functions.
$\bullet$ Problems with natural solutions and then generalizations
Like, for a binomial distribution(how they are giving the probability mass function).

 A: I would highly recommend A First Course in Probability by Sheldon Ross.
From the hyperlinked Amazon page:

A First Course in Probability features clear and intuitive explanations of the mathematics of probability theory, outstanding problem sets, and a variety of diverse examples and applications. This book is ideal for an upper-level undergraduate or graduate level introduction to probability for math, science, engineering and business students. It assumes a background in elementary calculus.

I am recommending this book because I think it is suitable for self-study, as it is listed as a textbook for the Society of Actuaries' Exam P.
You can also try checking out the following books:


*

*Mathematical Statistics with Applications by Dennis Wackerly, William Mendenhall and Richard L. Scheaffer

*Probability and Statistical Inference by Robert V. Hogg, Elliot Tanis and Dale Zimmerman

*Probability and Statistics with Applications: A Problem Solving Text by Leonard A. Asimow and Mark M. Maxwell
A: I think your best bet would be "Probability And Random Processes" by Grimmett and Stirzacker, see http://www.amazon.co.uk/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220. It provides a concise yet introductory treatment of probability theory, free of measure theory. The latter chapters may not be of use to you but the earlier chapters should be. This would be the quickest way to get the basics. 
I would also recommend, "A Course in Probability Theory" by Chung, http://www.amazon.co.uk/Course-Probability-Theory-Third-Edition/dp/0121741516, its a more rigorous treatment than others mentioned so far. If you are researching functional analysis, this should be quite accessible to you.
I would not recommend Ross' book at is it too elementary, too slow and not suitable for a postgraduate. It is aimed at first year mathematics students, many of whom are making the transition from high school. The only reason it is recommended is because when people who did one course in probability as a undergrad, this is the text that would have been recommended.
