Chance & Paradoxes

Probability measures how likely an event is, expressed as a value between zero and one.

Rolling two dice makes a total of seven the most likely outcome, since more combinations produce it.

The law of large numbers explains why outcomes settle near the expected average, and why casinos profit long-term.

Expected value predicts a random event's long-run average outcome, helping evaluate whether a bet is favorable.

The gambler's fallacy wrongly assumes past results affect future independent odds, expecting a losing streak to end.

Bayes' theorem updates event probabilities as new evidence arrives, used from medical testing to spam filters.

The Monty Hall problem shows that switching doors, though counterintuitive, actually improves your odds of winning.

The birthday paradox shows that just 23 people give roughly even odds that two share a birthday.

Simpson's paradox shows a data trend can reverse once separate groups are combined, a caution for statistics.

The two envelopes problem seems to always favor switching, exposing a flaw in naive expected-value reasoning.

The prisoner's dilemma shows individually rational choices can leave both players worse off, a founding game theory scenario.

Zeno's paradox argues motion is logically impossible by endlessly halving the distance left to travel.

The Ship of Theseus questions whether an object remains itself once every part has been replaced.

The twin paradox shows a traveling twin ages more slowly, a real effect predicted by Einstein's relativity.

Schrödinger's cat imagines a cat both alive and dead until observed, illustrating quantum mechanics' strangeness.

The grandfather paradox asks what happens if a time traveler prevents their own birth, challenging backward time travel.

Russell's paradox exposed a contradiction in naive set theory, forcing major revisions to mathematics' foundations.

The trolley problem forces a choice between differing casualty outcomes, widely used to explore moral reasoning.

The Sorites paradox, or paradox of the heap, questions exactly when removing grains stops something being a heap.

The liar paradox arises from a statement claiming to be false, which cannot be consistently true or false.

Random walks model paths built from random steps, used to describe stock prices and particle motion alike.

Combinatorics studies counting and arranging objects, providing the foundation for calculating probabilities.

Permutations and combinations count arrangements of objects, differing on whether order matters.