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choose a question, follow an idea

Where would you like to start?

Explore in any order. Each path explains why the next idea matters; the foundations are there whenever you need them.

Path 1 · 6 lessons

When intuition surprises you

Start with a familiar puzzle, then learn to ask what information and counting really mean.

Explore this path → Path 2 · 6 lessons

Counts and waiting times

Follow one process from a single success to counts, intervals, and accumulated waiting.

Explore this path → Path 3 · 6 lessons

Learning from samples

Move from a noisy sample to an estimate and an honest statement of uncertainty.

Explore this path →

Evidence, ambiguity, and decisions

Make sampling assumptions, priors, utilities, and causal comparisons explicit.

Sleeping Beauty

Compare trial sampling, awakening sampling, and betting schedules.

Ellsberg paradox

Inspect unknown composition and common-event cancellation.

Allais paradox

Change utilities and inspect the tickets shared by four lotteries.

Lindley's paradox

Hold z fixed while changing sample size and a proper alternative prior.

Newcomb's problem

Make the probability assumptions behind one-box and two-box comparisons explicit.

Doomsday argument

Compare uniform-rank evidence with population-weighted priors.

Information, records, and when to stop

Track what is known at each step and compare the value of continuing.

100 prisoners and hats

One parity bit makes every answer after the first certain.

Banach's matchbox

Count what remains when a pocket is first discovered empty.

Records in a permutation

Reveal record highs and derive their harmonic expected count.

Optimal house-selling

Compare each offer with the expected value of continuing.

Networks, reinforcement, and mixing

Change what gets sampled and how the current state shapes the next step.

Friendship paradox

Popular people appear more often when sampling friendship endpoints.

Parrondo's paradox

Switching games changes the capital states that determine their winning probabilities.

Riffle shuffle

Measure how a random cut-and-interleave shuffle approaches uniform permutations.

Pólya's urn

Reinforcement changes future draws while the expected urn fraction stays fixed.

Ehrenfest urn

Moving one ball flips parity; allowing a hold changes convergence.

Random walks in 1D, 2D, 3D

Compare finite return probabilities with recurrence in one and two dimensions and transience in three.

Kruskal count

Counting paths merge when they reach the same card; contact is not guaranteed.

Count paths, runs, and crowded bins

Move from independent trials to overlapping events and decisions that depend on current loads.

Galton board

Balls through pegs. The binomial builds itself as a bell in real time.

Longest run of heads

In n coin flips, how long is the longest streak? The answer grows like log₂ n.

Balls in bins

Throw n balls into n bins. Max load grows like ln n / ln ln n.

Power of two choices

Pick two bins at random, take the lesser-loaded. Max load collapses to ln ln n.

Make the sampling rule visible

Choose chords, narrow neighborhoods, and break sticks under explicit probability measures.

Bertrand paradox

Three natural samplers of a random chord give three different probabilities.

Borel's paradox

Conditioning on a measure-zero event depends on how you parameterize.

Broken stick

Break a stick twice at random. The pieces form a triangle exactly 1/4 of the time.

Choose the game before choosing a strategy

Compare head-to-head wins, pattern races, and matching sum distributions.

Non-transitive dice

Four fair dice form a cycle of favorable matchups. Change the scoring rule and compare.

Penney's game

Pick a coin-flip pattern; your opponent picks one that beats it. Non-transitive.

Sicherman dice

A non-standard pair of dice with the same sum distribution as 2d6.

Ask how the sample was selected

Change the reporting rule, group membership, or error costs and follow the denominators.

Boy or girl paradox

1/2 or 1/3? The answer depends on how you sampled the information.

Bertrand's box paradox

Gold and silver coins in three boxes. A crisper cousin of Monty Hall.

Berkson's paradox

Conditioning on admission makes independent traits look correlated.

Will Rogers phenomenon

Move an unchanged score between groups and watch both means rise.

Accuracy paradox

With 1% positives, a do-nothing classifier is 99% accurate. Why accuracy misleads.

Connect arrivals, variation, and prediction

After the foundations, follow one thread from event counts to uncertain rates.

Poisson process

Arrivals on a timeline. Gaps are exponential; counts in windows are Poisson.

Law of total variance

Separate within-group and between-group spread, then change the grouping.

Conjugate priors

Update beta and gamma priors, then integrate uncertainty into predictions.

From observed values to joint models

Build a sample distribution, model two measurements together, then learn a vector of category probabilities.

Empirical CDF

Turn observations into exact cumulative probabilities, tied-value masses, and inverse quantiles.

Multivariate Normal

Connect covariance, conditional distributions, linear projections, and perfect-correlation limits.

Dirichlet distribution

Learn category probabilities on a simplex and integrate their uncertainty into future counts.

When tails change the answer

Check what finite cutoffs and averaging can conceal about an unbounded distribution.

Zipf distribution

Compare infinite rank probabilities with a finite cutoff, and inspect tail mass and moment thresholds.

Symmetric stable distributions

See how normalized sums retain their shape while ordinary averages shrink, hold, or widen.

From uncertainty to prediction loss

Keep the same three outcomes as you move from entropy to cross-entropy and directional KL divergence.

Shannon entropy

Inspect expected surprise, code lengths, and the uncertainty of a finite source.

Cross-entropy

Separate source uncertainty from prediction error using expected log-loss.

KL divergence

Compare directional expected log-loss, signed contributions, and support failures.

Fit a model to observations

Maximum likelihood

Hold data fixed, vary the parameter, and check whether a maximum exists.

Follow a process through time

Brownian motion

Refine a Gaussian time grid while preserving the observations already drawn.

Martingales

Keep unfinished paths when checking whether a stopping rule preserves fairness.

Three ways intuition can miss the mechanism

Prisoners and boxes

Coordinate successes through permutation cycles.

Waiting for HH vs HT

Remember which suffix can start the next pattern.

Inspection paradox

Check whether you sampled intervals or points in time.

Choose and compare models

Check an arrival model with real data

Use a pinned earthquake catalog to compare count, timing, and rate assumptions.

How distributions connect

Follow transformations, special cases, mixtures, and limits with their conditions.

Binomial vs hypergeometric

Compare the same count with and without replacement from a finite population.

Which distribution fits the question?

Start with the observation and mechanism; identify assumptions still to check.

Poisson vs binomial

Hold the mean fixed and compare count probabilities under different mechanisms.

Normal vs Student’s t

Match location and scale, then measure how much the tails differ.

Build your foundations

Events and probability

Build a sample space, count outcomes, and take complements.

Conditioning and independence

New information changes the denominator. See which outcomes remain.

Reading a distribution

Connect probability mass, density, cumulative probability, and quantiles.

Expectation and variance

Locate the center, measure spread, and understand what an average promises.

Joint and marginal distributions

Read two variables together and separate within-group patterns from their mixture.