Paradoxes.

Switch or stay? Simulate the game and watch win rates diverge.

A 99% accurate test for a rare disease — what does a positive really mean?

A trend holds in every subgroup but reverses when pooled. Adjust case mixes to see it flip.

After five heads, tails must be due — right? Running proportions expose the error.

A game with infinite expected value — yet no one would pay much to play. Expected value breaks down.

Specify a prior and opening protocol to see why unconditional symmetry does not settle every conditional decision.

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

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

Compare trial sampling, awakening sampling, and betting schedules.

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

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

Compare uniform-time and uniform-interval observations of the same schedule.

Popular people appear more often when sampling friendship endpoints.

Conditioning on admission makes independent traits look correlated.

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

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

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

Inspect unknown composition and common-event cancellation.

Change utilities and inspect the tickets shared by four lotteries.

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

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

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

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

Compare uniform-rank evidence with population-weighted priors.