Introduction to Probability

Video 2

sample space, union, intersection, complement, set difference, De Morgan’s laws, disjoint events and partition

Builds on Probability Models and Uncertainty.

Video 3

inclusive or, neither, not both, exactly one, at least, at most, only if and counterexample

Builds on Sample Spaces and Events.

Video 4

probability axioms, nonnegativity, normalization, countable additivity, pairwise disjoint events, probability of the empty event and monotonicity

Builds on Sample Spaces and Events.

Video 5

complement rule, complementary partition, at least one, not all and complement of at least k

Builds on Probability Axioms and Basic Rules and Translating Event Language.

Video 6

addition rule, overlap, disjoint regions, exactly one of two events and union bound

Builds on Probability Axioms and Basic Rules and Translating Event Language.

Video 7

equally likely outcomes, uniform finite model, favorable outcomes, cardinality, level of detail and ordered sequences

Builds on Probability Axioms and Basic Rules, The Complement Rule and The Addition Rule.

Video 8

simulation, simulated trial, pseudorandom generator, random seed, reproducibility, running estimate and sampling variability

Builds on Equally Likely Outcomes.

Video 9

product rule for counting, sum rule for counting, counting by cases, selection with replacement, selection without replacement, counting with restrictions and decision tree

Builds on Equally Likely Outcomes.

Video 10

factorial, zero factorial, permutation, partial permutation and circular arrangement

Builds on Sum and Product Rules for Counting.

Video 11

combination, binomial coefficient, overcounting, symmetry of binomial coefficients, Pascal’s identity, Pascal’s triangle and binomial theorem

Builds on Permutations.

Video 12

multiset permutation, indistinguishable objects, multinomial coefficient, counting by choosing positions, allocation to named groups and multinomial expansion

Builds on Combinations and Binomial Coefficients.

Video 13

inclusion-exclusion principle, pairwise intersection, triple intersection, disjoint Venn regions, exactly one of three events and exactly two of three events

Builds on The Addition Rule and Combinations and Binomial Coefficients.

Video 14

collision, no-collision probability, birthday problem, match of a specified pair, pigeonhole principle and nonuniform categories

Builds on Inclusion-Exclusion and The Complement Rule.

Video 15

ordered selection, unordered selection, unordered selection with repetition, multiplicity of a record, equal likelihood after erasing order and counting by complement

Builds on Sum and Product Rules for Counting, Permutations, Combinations and Binomial Coefficients, Repeated Objects and Multinomial Counting, Inclusion-Exclusion and Birthday and Collision Problems.

Video 16

conditional probability, conditioning event, reference group, reversed condition and positive-probability requirement

Builds on Probability Axioms and Basic Rules and The Addition Rule.

Video 17

two-way table, joint probability, marginal probability, margins, row and column percentages and natural frequencies

Builds on Conditional Probability.

Video 18

independent events, product criterion, dependent events, independence of complements, pairwise independence and mutual independence

Builds on Conditional Probability.

Video 19

disjoint events with positive probability, overlapping independent events, overlapping dependent events, zero-probability event and nested events

Builds on Independent Events and The Addition Rule.

Video 20

probability tree, conditional branch, multiplication rule, chain rule, terminal path and branch normalization

Builds on Sum and Product Rules for Counting, Conditional Probability and Independent Events.

Video 21

law of total probability, group weight, conditional rate, weighted average and zero-weight group

Builds on Conditional Probability and Probability Trees and the Chain Rule.

Video 22

Bayes’ theorem, prior probability, likelihood, posterior probability, base rate neglect, sensitivity, specificity and false positive rate

Builds on The Law of Total Probability.

Video 23

conditional independence, latent state, shared cause, marginal dependence, sequential updating and duplicated evidence

Builds on Bayes’ Theorem and Base Rates and Independent Events.

Video 24

odds in favor, odds against, probability from odds, likelihood ratio, prior odds, posterior odds and odds form of Bayes’ theorem

Builds on Bayes’ Theorem and Base Rates and Conditional Independence and Sequential Evidence.

Video 25

random variable, realization, discrete random variable, support, countably infinite support and distribution of a random variable

Builds on Equally Likely Outcomes, Conditional Probability and Independent Events.

Video 26

probability mass function, nonnegative masses, normalizing constant, event probability as a sum of masses, finite uniform distribution and probability bar chart

Builds on Discrete Random Variables.

Video 27

cumulative distribution function, step function, jump size, right-continuity, tail probability and interval probability

Builds on Probability Mass Functions.

Video 28

function of a random variable, preimage, one-to-one map, many-to-one map, collecting probability mass and positive part

Builds on Probability Mass Functions and Cumulative Distribution Functions.

Video 29

expected value, probability-weighted average, mode, conditional mean, net gain, infinite expectation and absolute integrability

Builds on Discrete Random Variables and Probability Mass Functions.

Video 30

expected payoff, net payoff, fair price, break-even probability, expected cost, model risk and decision criterion

Builds on Expected Value.

Video 31

linearity of expectation, expectation of a sum, linear combination, expectation of a product, expectation of a square and finite expectation

Builds on Expected Value and Functions of Discrete Random Variables.

Video 32

indicator random variable, expectation of an indicator, sum of indicators, expected count, indicator of a union and expected number of distinct categories

Builds on Linearity of Expectation and Discrete Random Variables.

Video 33

variance, deviation from the mean, squared deviation, second moment, computational formula for variance, squared units and finite second moment

Builds on Expected Value, Linearity of Expectation and Functions of Discrete Random Variables.

Video 34

standard deviation, root-mean-square deviation, mean absolute deviation, shift, change of scale, affine transformation and change of units

Builds on Variance.

Video 35

distribution of a sum, ordered-pair table, discrete convolution, variance of an independent sum and cross term

Builds on Independent Events, Probability Mass Functions, Linearity of Expectation and Variance.

Video 36

random sample, sample mean, statistic, sampling distribution, independent and identically distributed and standard error

Builds on Sums of Discrete Random Variables and Cumulative Distribution Functions.

Video 37

discrete uniform distribution, number of support values, mean by symmetry, variance of consecutive integers, sum-of-squares identity and evenly spaced support

Builds on Probability Mass Functions, Expected Value and Variance. Also uses the sum of the first n positive integers.

Video 38

Bernoulli distribution, Bernoulli trial, success, parameter of a distribution, mean and variance of a Bernoulli variable, degenerate distribution and maximum Bernoulli variance

Builds on Probability Mass Functions, Expected Value and Variance.

Video 39

binomial distribution, binomial probability formula, binomial assumptions, mean of a binomial count, variance of a binomial count and binomial tail probability

Builds on The Bernoulli Distribution, Combinations and Binomial Coefficients, Variance, Sums of Discrete Random Variables and Indicator Random Variables.

Video 40

geometric distribution, trial-count convention, failure-count convention, geometric series, geometric tail, memoryless property and mean and variance of a geometric distribution

Builds on The Bernoulli Distribution and The Binomial Distribution.

Video 41

negative binomial distribution, stopping rule, waiting time for the rth success, failure-count form, sum of geometric waiting times and deadline probability as a binomial tail

Builds on The Geometric Distribution, The Binomial Distribution and Combinations and Binomial Coefficients.

Video 42

hypergeometric distribution, finite population, sampling without replacement, support of a hypergeometric count, finite population correction and dependent draws

Builds on Combinations and Binomial Coefficients, Probability Trees and the Chain Rule and The Binomial Distribution.

Video 43

Poisson distribution, Poisson parameter, rate times exposure, exponential series, Poisson mean and variance and Poisson approximation to the binomial

Builds on The Binomial Distribution, Expected Value and Variance. Also uses the exponential function and the number e.

Video 44

random vector, joint probability mass function, joint probability table, joint support, event region and function of a random pair

Builds on Probability Mass Functions and Conditional Probability.

Video 45

marginal distribution, marginal probability mass function, summing out a variable and information lost by marginalizing

Builds on Joint Probability Mass Functions.

Video 46

conditional probability mass function, renormalizing a row or column, reversed conditioning, conditional mean from a conditional PMF and joint probability from conditional and marginal

Builds on Marginal Probability Distributions, Conditional Probability and Expected Value.

Video 47

independent random variables, product law, joint cumulative distribution function, CDF factorization, functions of independent variables and mutual independence of random variables

Builds on Independent Events, Joint Probability Mass Functions, Marginal Probability Distributions, Conditional Probability Mass Functions and Cumulative Distribution Functions.

Video 48

covariance, correlation coefficient, uncorrelated variables, variance of a sum, bilinearity of covariance, Cauchy-Schwarz inequality and perfect correlation

Builds on Variance, Independent Random Variables and Joint Probability Mass Functions.

Video 49

conditional expectation, conditional mean function, random conditional mean, law of total expectation, integrability and squared-error prediction

Builds on Expected Value, Linearity of Expectation and Conditional Probability Mass Functions.

Video 50

conditional variance, law of total variance, within-group variation and between-group variation

Builds on Variance, Covariance, Correlation and Sums and Conditional Expectation and the Tower Rule.

Video 51

random sum, random count, empty sum, mean of a random sum and variance of a random sum

Builds on The Law of Total Variance, Poisson Distributions and Linearity of Expectation.

Video 52

realized average, unbiasedness, variance of the sample mean, standard error of the mean and square-root scaling

Builds on Sample Means and Sampling Distributions, Independent Random Variables and Covariance, Correlation and Sums.

Video 53

probability bound, Markov’s inequality, Chebyshev’s inequality, sharp bound, two-sided deviation event and bound in standard deviation units

Builds on Expected Value, Variance and Mean and Variance of the Sample Mean.

Video 54

weak law of large numbers, convergence in probability, error tolerance, relative frequency, sufficient sample size and finite-sample uncertainty

Builds on Markov and Chebyshev Inequalities, Mean and Variance of the Sample Mean and The Binomial Distribution.

Video 55

streak, gambler’s fallacy, dilution, compensation misconception, run at a preselected position and learning about an unknown probability

Builds on Law of Large Numbers for Discrete Variables and Independent Events.

Video 56

simulation estimate, Monte Carlo error, plug-in standard error, conservative standard error, replication, model error and implementation error

Builds on Probability Simulation, Mean and Variance of the Sample Mean and Law of Large Numbers for Discrete Variables.

Video 57

repeated sampling, exact benchmark, empirical distribution, empirical variance and replication count

Builds on Simulation Error and Number of Trials, The Binomial Distribution and Law of Large Numbers for Discrete Variables.

Video 58

probability density, probability as area, Riemann sum, definite integral, limits of integration, accumulated probability and power rule for integrals

Builds on Discrete Random Variables, Expected Value and Variance. Also uses areas of rectangles and triangles and an informal idea of a limit.

Video 59

difference quotient, derivative, tangent slope, power rule for derivatives, fundamental theorem of calculus, antiderivative and density as the derivative of the CDF

Builds on Integrals and Probability as Area and Cumulative Distribution Functions.

Video 60

probability density function, absolutely continuous distribution, normalization of a density, density units, CDF of a continuous variable, interval probability from a CDF and zero probability at a point

Builds on Integrals and Probability as Area and Derivatives and the Density-CDF Connection.

Video 61

point mass, mixed distribution, CDF jump, left limit of a CDF, probability-zero event and almost surely

Builds on Probability Densities and CDFs and Cumulative Distribution Functions.

Video 62

expected value of a continuous variable, law of the unconscious statistician, variance of a continuous variable, mean of a mixed distribution, infinite mean and moment existence

Builds on Integrals and Probability as Area, Derivatives and the Density-CDF Connection, Probability Densities and CDFs and Variance. Also uses integrals over an unbounded interval.

Video 63

continuous uniform distribution, quantile, percentile, median, mean and variance of a uniform distribution, conditional uniform distribution and change of units for a density

Builds on Probability Densities and CDFs and Continuous Expectation and Variance.

Video 64

exponential distribution, rate, scale parameter, survival function, memorylessness, natural logarithm and exponential quantile

Builds on The Geometric Distribution, Probability Densities and CDFs, Continuous Expectation and Variance and Uniform Distributions. Also uses derivatives of exponential functions and the product rule.

Video 65

normal distribution, normal density, standard normal distribution, standardization, 68-95-99.7 rule and normal tail probability

Builds on Probability Densities and CDFs and Continuous Expectation and Variance. Also uses using a standard normal table or calculator.

Video 66

z-score, standard normal CDF, symmetry of the standard normal CDF, upper-tail probability, normal quantile, inverse CDF and central probability interval

Builds on The Normal Distribution.

Video 67

distribution family, parameter convention, support of a model, tail behavior and model adequacy

Builds on Uniform Distributions, Exponential Distributions, The Normal Distribution and Normal Probabilities and Quantiles.

Video 68

CDF method, monotone transformation, many-to-one transformation, unbounded density, inverse-CDF sampling and generalized quantile

Builds on Probability Densities and CDFs, Uniform Distributions, Exponential Distributions and Cumulative Distribution Functions.

Video 69

weak law of large numbers for continuous variables, running average, Chebyshev bound for a sample mean, conservative bound and perfectly dependent observations

Builds on Law of Large Numbers for Discrete Variables, Continuous Expectation and Variance and Mean and Variance of the Sample Mean.

Video 70

central limit theorem, standardized sum, convergence in distribution, normal approximation for an average, finite positive variance and heavy tails

Builds on Exponential Distributions, The Normal Distribution, Mean and Variance of the Sample Mean and The Law of Large Numbers for Continuous Variables.

Video 71

rare-event regime, normal approximation to the binomial, continuity correction, absolute error, relative error and tail accuracy

Builds on The Binomial Distribution, Poisson Distributions, The Normal Distribution and The Central Limit Theorem.

Video 72

Monte Carlo method, Monte Carlo estimator, Monte Carlo standard error, probability as an expected indicator, square-root cost of precision and approximate interval for a Monte Carlo estimate

Builds on Probability Simulation, Transformations and Inverse-CDF Sampling, Simulation Error and Number of Trials and The Central Limit Theorem.

Video 73

exact model probability, approximation error, simulation error, uncertain model inputs and error interpretation

Builds on Poisson and Normal Approximations and Monte Carlo Probability and Expectation.

Video 74

random process, time index, Bernoulli process, cumulative count, first success time, disjoint blocks of trials and count and waiting time events

Builds on The Bernoulli Distribution, The Geometric Distribution and The Binomial Distribution.

Video 75

simple random walk, increment, position, drift, variance of a random walk, parity restriction and path probability

Builds on Bernoulli Processes, The Binomial Distribution and Covariance, Correlation and Sums.

Video 76

counting process, homogeneous Poisson process, rate of arrivals, independent increments, stationary increments and overlapping counts

Builds on Poisson Distributions, Exponential Distributions and Bernoulli Processes.

Video 77

first arrival time, exponential waiting time, interarrival time, kth arrival time, count and waiting time connection and memoryless waiting

Builds on Poisson Process Basics and Exponential Distributions.

Video 78

Markov chain, state, Markov property, time-homogeneous chain, transition matrix, row-stochastic convention, initial distribution and one-step update

Builds on Conditional Probability, Conditional Independence and Sequential Evidence and Bernoulli Processes. Also uses matrix multiplication.

Video 79

n-step transition probability, matrix power, row vector of state probabilities, Chapman-Kolmogorov relation, endpoint probability and visiting probability

Builds on Markov Chains and Transition Matrices.

Video 80

stationary distribution, balance equations, probability flow, convergence of the state distribution, periodic chain and nonunique stationary distribution

Builds on Multi-Step Markov Probabilities.

Video 81

hitting time, hitting probability, absorbing state, first-step analysis, boundary condition, expected time to absorption and infinite hitting time

Builds on Conditional Expectation and the Tower Rule, Markov Chains and Transition Matrices and Stationary Distributions.

Video 82

state definition, Markov assumption, refined state, empirical transition matrix, time step, checking time homogeneity and model limitation

Builds on Stationary Distributions and Hitting Times and Absorption.

Video 83

model specification, joint model from conditional rates, binomial count of alerts, conditional expected work, assumption sensitivity and prediction versus simulated output

Builds on Bayes’ Theorem and Base Rates, The Binomial Distribution, Conditional Expectation and the Tower Rule, Law of Large Numbers for Discrete Variables and Simulation Error and Number of Trials.

Video 84

combined probability model, independence of submodels, fixed batch versus fixed time window, checkpoint availability, sensitivity analysis and simulation check of a submodel

Builds on A Discrete Probability Investigation, Uniform Distributions, Exponential Distributions, The Normal Distribution, Monte Carlo Probability and Expectation, Poisson Process Basics, Arrival Counts and Waiting Times, Stationary Distributions, Hitting Times and Absorption and Choosing and Checking Markov Models.