Stanford CS109 Lecture 1 | What is Probability?: List outcomes first; only then assign probabilities to events.
List outcomes first; only then assign probabilities to events.
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23 postsStanford CS109 Lecture 2 | Conditional Probability: A condition restricts the sample space to outcomes still compatible with the evidence.
A condition restricts the sample space to outcomes still compatible with the evidence.
Stanford CS109 Lecture 3 | Bayes Theorem: Bayes’ theorem turns an easier generative direction into the inferential direction we need.
Bayes’ theorem turns an easier generative direction into the inferential direction we need.
Stanford CS109 Lecture 4 | Counting and Combinatorics: Decide whether order matters and repetition is allowed before choosing a formula.
Decide whether order matters and repetition is allowed before choosing a formula.
Stanford CS109 Lecture 5 | Random Variables and Expectation: A random variable maps outcomes to numbers; expectation is a weighted average, not necessarily an attainable value.
A random variable maps outcomes to numbers; expectation is a weighted average, not necessarily an attainable value.
Stanford CS109 Lecture 6 | Moments: Expectation, LOTUS, and linearity
Expectation compresses a distribution into a weighted average; LOTUS handles transformed values, while linearity makes sums tractable even without independence.
Stanford CS109 Lecture 7 | Variance and Poisson: From spread to rare-event counts
Variance describes a random variable's spread; Poisson models counts in a fixed interval and approximates a large-n, small-p binomial.
Stanford CS109 Lecture 8 | Continuous Random Variables: PDFs, CDFs, Uniform, and Exponential
A continuous variable assigns zero probability to a point and area to intervals; CDFs, Uniform, Exponential, and memorylessness build on that distinction.
Stanford CS109 Lecture 9 | Normal Distribution: Standardization, Phi, and continuity correction
Standardization maps Normal variables to Z; Phi, linear transforms, and continuity correction turn intervals and large binomials into computable probabilities.
Stanford CS109 Lecture 10 | Probabilistic Models: Joints, marginals, independence, and Bayes
A joint distribution retains the full relationship among variables; marginals, conditionals, independence, and Bayes extract different answers from it.
Stanford CS109 Lecture 11 | Inference: Prior times likelihood, then normalize
Inference multiplies each hidden-variable prior by an observation likelihood and normalizes; the same loop handles repeated evidence and discretized continuous beliefs.
Stanford CS109 Lecture 12 | General Inference: Bayesian networks, sampling, and rare evidence
A Bayesian network factorizes a huge joint through conditional independence; ancestral sampling generates joint samples, and rejection sampling filters them into a conditional.
Stanford CS109 Lecture 13 | Multinomial: Category counts, bag of words, and log probability
The Multinomial extends two-category Binomial counts to many categories; the same PMF models documents as word counts for Bayesian authorship with log-scores.
Stanford CS109 Lecture 14 | Beta: Turn an unknown probability into an updatable random variable
A Beta distribution represents full belief about an unknown success rate; success/failure data updates two parameters for posteriors, smoothing, and Thompson-sampling decisions.
Stanford CS109 Lecture 15 | Adding Random Variables and the Central Limit Theorem
A few independent sums have closed forms; general IID sums become approximately Normal under the CLT, with continuity correction for discrete sums.
Stanford CS109 Lecture 16 | Bootstrapping: Sampling statistics, error bars, and p-values
The bootstrap treats a sample histogram as a population proxy, resampling with replacement to approximate a statistic's sampling distribution, error bar, or null p-value.
Stanford CS109 Lecture 17 | Algorithmic Analysis: Conditional expectation, indicators, and recursion
Expected cost in randomized code can be conditioned on the first random choice; counting problems become indicator sums, often avoiding the full distribution entirely.
Stanford CS109 Lecture 18 | Information Theory: Surprise, entropy, information gain, and KL
Surprise turns rare events into bits; entropy is expected surprise, information gain selects uncertainty-reducing questions, and KL measures excess cost from a model distribution.
Stanford CS109 Lecture 19 | Maximum Likelihood Estimation: Hold data fixed and optimize the parameter
MLE fixes observed data and optimizes parameters; log-likelihood turns products into sums, but a maximum can also lie on a boundary.
Stanford CS109 Lecture 20 | Logistic Regression: Derive the gradient from Bernoulli likelihood
Logistic regression turns a linear score into a Bernoulli probability with sigmoid; the gradient xⱼ(y-ŷ) follows directly from the log-likelihood chain rule.
Stanford CS109 Lecture 21 | Comparing Classifiers: Beyond accuracy to calibration, error costs, and fairness
Classifier comparison requires held-out data, baselines, calibration, precision/recall, and an explicit fairness criterion—not accuracy alone.
Stanford CS109 Lecture 22 | Deep Learning: Derive backpropagation with the chain rule
A neural network stacks logistic units; a forward pass computes probabilities, while backpropagation reuses output error to obtain every gradient.
Stanford CS109: A Probability Course That Turned "How to Read This Lecture With an LLM" Into Official Coursework
Every lecture in CS109's Summer 2026 offering ships with an official LLM Learning Guide — six concepts, a Learn prompt and a Test me prompt for each, written week by week across the quarter for a total of 23 PDFs. The same course's honor code Rule 4 forbids asking an LLM to solve your homework, and 65% of the grade sits in proctored exam rooms. Those two facts are halves of one design.