Intuitive concepts: probability of an event as a measure between 0 and 1; random variable; probability distribution; frequency interpretation of probability; random numbers; coins, dice, and other games; simulations; odds; historical development of probability; random walks.

Formal concepts: sample space, outcomes, and events; random variable; discrete distribution functions and axioms of probability; unions, intersections, and complements; properties of probabilities, principle of inclusion and exclusion; tree diagrams; uniform distributions over finite sets, symmetry; infinite sample spaces with discrete probabilities.

The intuitive problems with probabilities over space (line, plane, Rⁿ in general). Monte Carlo simulations, Buffon's needle.

Formal concepts: density function for a continuous random variable; integration; cumulative distribution functions; derivatives; exponential density function;

Counting problems. Permuations; tree diagrams; combinations, binomial coefficients, binomial theorem, and Pascal's triangle; Bernoulli trials, Bernoulli probabilities, binomial distributions; hypothesis testing; general principle of inclusion and exclusion.

Intutive concept of conditional probability; formal definition of conditional probability; Bayes' formula for inverting conditional probabilities; independent events; joint distribution functions; independent random variables; independent trials. Conditional density functions for continuous distributions; the beta distribution

Uniform continuous distributions; geometric distribution; Poisson distribution; exponential and gamma distributions; introduction to queueing theory; normal (Gaussian) distribution; Chi-squared distribution

Expected value for discrete random variables, expectation; linearity of expectation; expectation of independent random variables; conditional expectation; variance and standard deviation; variance of various distributions. Expectation and variance for continuous random variables.

Analysis of sums of independent random variables with identical distributions, that is, independent trials.

Chebychev inequality, law of averages, law of large numbers.

The central limit theorem for Bernoulli trials, binomial distributions again, the normal distribution, the general central limit theorem.

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