Mathematical set-up for probability, equally likely outcomes, basic counting arguments, conditional probability, Bayes' rule, independent events.
Sampling with and without replacement, Binomial distribution, Normal and Poisson approximations.
Discrete random variables, Binomial, Poisson, Hypergeometric, Geometric distributions, expectation, variance, generating functions.
Discrete joint distributions, independent variables, repeated trials.
Law of Large Numbers, Central Limit Theorem.
Continuous random variables, density, cumulative distribution function, change of variable, expectation, variance.
Uniform, Exponential, Gamma, Normal distribution.
Poisson process.
Continuous joint distribution, independence.
Discrete and continuous conditional distributions and conditional expectations.
Bivariate Normal distribution.

Calculus ( Math - 53 or equivalent )

Probability by Jim Pitman.