Sets · Venn diagrams · Counting
An interactive introduction to probability and statistics
Chance,
Data, and Decisions
Start with what might happen. Observe what did. Then learn how evidence turns uncertainty into a calibrated decision.
model
→simulate
→observe
→infer
→check
The course
One uncertainty unlocks the next.
The sequence follows a single intellectual arc: outcomes become variables, variables form distributions, distributions generate samples, and samples support inference.
Spaces · Events · Probability axioms
A measure of possibility
Build coherent models from outcomes and events.Conditioning · Independence · Bayes
Information changes the space
Restrict, renormalize, and reverse the direction of evidence.Random variables · CDFs · Moments
Numbers attached to outcomes
Compress experiments into quantities we can analyze.Binomial · Poisson · Normal · Beta
A library of chance mechanisms
Recognize the recurring mechanisms behind familiar laws.Joint laws · Marginals · Correlation
Variables together
See dependence as a whole surface, not a single number.LLN · CLT · Sampling distributions
From repetition to regularity
Watch averages stabilize and bell curves emerge.Estimation · Intervals · Tests
Learning from a sample
Turn random data into calibrated claims.Prior · Likelihood · Posterior
Probability for unknowns
Update uncertainty and predict what comes next.Designed for a first undergraduate courseTwo 75-minute meetings per week · algebra and introductory calculus
Download the complete 30-page packetInteractive companion
Make randomness visible.
Every lab pairs a manipulable model with an interpretation question. Simulation is a microscope for assumptions—not a substitute for them.
Event lab
Shade the statement
Switch between words, symbols, and regions. Probability begins with getting the event right.
Read the region
A ∩ B: both events occurABincluded?0 · 0 no0 · 1 no1 · 0 no1 · 1 yes
Before computing, ask: are the outcomes ordered? Are they equally likely? Is the event inclusive or exclusive?
Bayes lab
Reverse the condition
A positive result is evidence, not a verdict. Watch the base rate reshape the answer.
true positivefalse positiveother
Among positive results
8.8%actually have the condition10/10 + 99true positives / all positives
Distribution lab
Shape a probability law
Change the mechanism, then read its center and spread before looking at the graph.
mean4.20
variance2.73
Count successes in a fixed number of independent trials.
Dependence lab
Turn the correlation dial
Correlation captures linear alignment—not every form of dependence, and never causation by itself.
ρ
0.70moderate linear associationZero correlation only means no linear trend. A curve can still be perfectly predictable.
Sampling lab
Repeat the sample, not the slogan
See the CLT reshape averages, then watch confidence intervals succeed and occasionally miss.
2,600 simulated sample means
center 1.00standard error 0.447
28 nominal 95% intervals · 27 cover μ
Bayesian lab
Watch evidence move a prior
Treat uncertainty about a proportion as a distribution, update it, then predict the next trial.
prior Beta(2,2)posterior Beta(10,4)
prior mean0.500
+data8/10
→posterior mean0.714
Posterior predictive chance of success on the next trial: 71.4%
Practice architecture
Calculate. Explain. Critique.
A complete assignment mixes technical fluency, mathematical reasoning, and model criticism.
Translate precisely
Move among words, events, tables, distributions, and formulas.
Example promptFind the CDF, then recover every point probability from its jumps.
Justify the bridge
Derive results and identify exactly where assumptions enter.
Example promptProve the weak law from Chebyshev’s inequality.
Stress-test a model
Simulate, visualize, compare, and diagnose a mismatch.
Example promptWhen does a Poisson approximation become convincing—and by which metric?
Computational companion
Four ready-to-run Python notebooks
The governing question
What would we expect to see if our model were true?
That question joins probability, simulation, estimation, testing, and posterior prediction. It also keeps inference honest.
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