- Counting and combinatorics
- Sets: unions, intersections, partitions
- De Morgan's Laws
- The Inclusion-Exclusion principle
- The Fundamental Rule of Counting
- Combinations
- Permutations
- Strategies for counting

- Theories of probability
- Equally likely outcomes
- Frequency Theory
- Subjective Theory
- Shortcomings of the theories
- Rates versus probabilities
- Measurement error
- Where does probability come from in physical problems?
- Making sense of geophysical probabilities
- Earthquake probabilities
- Probability of magnetic reversals
- Probability that Earth is more than 5B years old

- Axiomatic Probability
- Outcome space and events, events as sets
- Kolmogorov's axioms (finite and countable)
- Analogies between probability and area or mass
- Consequences of the axioms
- Probabilities of unions and intersections
- Bounds on probabilities
- Bonferroni's inequality
- The inclusion-exclusion rule for probabilities

- Conditional probability
- The Multiplication Rule
- Independence
- Bayes Rule

- Random variables.
- Probability distributions of real-valued random variables
- Cumulative distribution functions
- Discrete random variables
- Probability mass functions
- The uniform distribution on a finite set
- Bernoulli random variables
- Random variables derived from the Bernoulli
- Binomial random variables
- Geometric
- Negative binomial

- Hypergeometric random variables
- Poisson random variables: countably infinite outcome spaces

- Random variables, continued
- Continuous and "mixed" random variables
- Probability densities
- The uniform distribution on an interval
- The Gaussian distribution

- The CDF of discrete, continuous, and mixed distributions
- Distribution of measurement errors
- The box model for random error
- Systematic and stochastic error

- Independence of random variables
- Events derived from random variables
- Definitions of independence
- Independence and "informativeness"
- Examples of independent and dependent random variables
- IID random variables
- Exchangeability of random variables

- Marginal distributions
- Point processes
- Poisson processes
- Homogeneous and inhomogeneous Poisson processes
- Spatially heterogeneous, temporally homogenous Poisson processes as a model for seismicity
- The conditional distribution of Poisson processes given N

- Marked point processes
- Inter-arrival times and inter-arrival distributions
- Branching processes
- ETAS

- Poisson processes

- Expectation
- The Law of Large Numbers
- The Expected Value
- Expected value of a discrete univariate distribution
- Special cases: Bernoulli, Binomial, Geometric, Hypergeometric, Poisson

- Expected value of a continuous univariate distribution
- Special cases: uniform, exponential, normal

- Expected value of a multivariate distribution

- Expected value of a discrete univariate distribution
- Standard Error and Variance.
- Discrete examples
- Continuous examples
- The square-root law
- Standardization and Studentization
- The Central Limit Theorem

- The tail-sum formula for the expected value
- Conditional expectation
- The conditional expectation is a random variable
- The expectation of the conditional expectation is the unconditional expectation

- Useful probability inequalities
- Markov's Inequality
- Chebychev's Inequality
- Hoeffding's Inequality
- Jensen's inequality

- Simulation
- Pseudo-random number generation
- Importance of the PRNG. Period, DIEHARD

- Assumptions
- Uncertainties
- Sampling distributions

- Pseudo-random number generation

- Hypothesis tests
- Null and alternative hypotheses, "omnibus" hypotheses
- Type I and Type II errors
- Significance level and power
- Approximate, exact, and conservative tests
- Families of tests
- P-values
- Estimating P-values by simulation

- Test statistics
- Selecting a test statistic
- The null distribution of a test statistic
- One-sided and two-sided tests

- Null hypotheses involving actual, hypothetical, and counterfactual randomness
- Multiplicity
- Per-comparison error rate (PCER)
- Familywise error rate (FWER)
- The False Discovery Rate (FDR)

- Tests, continued
- Parametric and nonparametric tests
- The Kolmogorov-Smirnov test and the MDKW inequality
- Example: Testing for uniformity
- Conditional test for Poisson behavior

- Permutation and randomization tests
- Invariances of distributions
- Exchangeability
- The permutation distribution of test statistics
- Approximating permutation distributions by simulation
- The two-sample problem

- Testing when there are nuisance parameters

- Parametric and nonparametric tests
- Confidence sets
- Definition
- Interpretation
- Duality between hypothesis tests and confidence sets
- Tests and confidence sets for Binomial p
- Pivoting
- Confidence sets for a normal mean
- known variance
- unknown variance; Student's t distribution

- Confidence sets for a normal mean
- Approximate confidence intervals using the normal approximation
- Empirical coverage
- Failures

- Nonparametric confidence bounds for the mean of a nonnegative population
- Multiplicity
- Simultaneous coverage
- Selective coverage