1. Introduction: empirical background, sample space, events, probability as a set function. Enumeration problems, inclusion-exclusion formula, urn models, problems of geometric origin. 

2. Conditional probability: basic concepts, multiplication rule, law of total probability, Bayes formula, applications. Independence. 

3. Discrete random variables: probability mass function, Bernoulli, geometric, binomial, hypergeometric and negative binomial distributions. Poisson approximation of the binomial distribution, Poisson distribution, Poisson process, applications. 

4. General theory of random variables: (cumulative) distribution function and its properties, singular continuous distributions, absolutely continuous distributions and probability density functions. Important continuous distributions: uniform, exponential, normal (Gauss), Cauchy. Distribution of a function of a random variable, transformation of probability densities. 

5. Quantities associated to distributions: expected value, moments, median, variance and their properties. Computation for the important distributions. Steiner formula. Applications. 

6. Joint distributions: joint distibution, mass and density functions, marginal and conditional distributions. Important joint distributions: polynomial, uniform and mutlidimensional normal distribution. Sums of independent variables, convolution, Conditional distribution and density functions. Conditional expectation and prediction, tower rule, conditional variance. Vector of expected values, Covariance matrix, Cauchy-Schwartz inequality, correlation. Indicator random variables. 

7. Weak Law of Large Numbers: Bernoulli’s Law of Large Numbers, Markov and Chebyshev inequality. Weak Law of Large numbers in full generality. 

8. Normal approximation of binomial distribution: Stirling formula, de Moivre-Laplace theorem. Applications. Normal fluctuations. Central Limit Theorem.