Probability Paradox Lab

Eight new paradoxes — 24 themes in all. Survivorship Bias, Buffon's Needle, Sleeping Beauty and more. 72 quiz questions. Now in 19 languages.

"With 23 people in a class, what's the chance two share a birthday?"

The answer exceeds 50%. Were you surprised?

Probability Paradox Lab is an interactive app that lets you experience mathematical paradoxes through simulation — results that feel wrong, yet are mathematically proven.

[ HOW IT WORKS ]

1. Make your intuitive guess
2. Run the simulation and see the real result
3. Read the math explanation and understand why

Be surprised by the answer. Be surprised again by the explanation.

[ FREE THEMES ]

Birthday Problem (Free) With just 23 people, the probability of a shared birthday exceeds 50%. Why?

Monty Hall Problem (Free) After the host opens a losing door, switching wins 2/3 of the time. Mathematically proven — yet deeply counterintuitive.

Two Envelopes Problem (Free) A seemingly valid argument says you should always switch envelopes. Where's the flaw?

Coupon Collector Problem (Free) Collecting all 10 kinds takes about 29 pulls on average. Fast at first, then it stalls.

[ PREMIUM THEMES ]

One-time purchase unlocks 20 more paradoxes and Quiz Mode across all 24 themes (72 questions). No subscriptions, no ads for premium users.

Ellsberg Paradox Why do people prefer known odds over unknown ones — even when the math says they're equal?

Boy or Girl Paradox "One child is a boy." How does that single fact change the probability?

False Positive Paradox A 99%-accurate test says positive for a disease 1% of people have. You're sick with only ~50% odds.

Gambler's Fallacy Five heads in a row. Is tails due? Randomness is clumpier than it looks.

St. Petersburg Paradox Expected value is infinite, yet you'd pay little to play. Why?

Nontransitive Dice (Efron-style) A beats B, B beats C, C beats A. No die is the strongest.

Parrondo's Paradox Two losing games, played in alternation, somehow become a winning strategy.

Simpson's Paradox A treatment wins in every subgroup — yet loses overall. How?

Secretary Problem Interview one at a time with no going back. The optimal strategy for catching the best.

Benford's Law About 30% of naturally occurring numbers start with a 1. A curious statistical bias.

Friendship Paradox Your friends have more friends than you — the network bias that makes it true.

100 Prisoners Problem If even one fails, all are executed. Yet one strategy saves everyone about 31% of the time.

Survivorship Bias Bullet holes on bombers that came back mark where a plane can be hit and still fly home. Armor the untouched parts.

Regression to the Mean Praise and they slip, scold and they improve — not the coaching, just probability swinging back.

Gambler's Ruin Even on a perfectly fair coin, the side with the smaller bankroll goes broke almost every time.

Buffon's Needle Drop needles on a lined floor and you can measure π.

Pólya's Urn Return the color you drew plus one more, and early luck gets locked in forever.

The Arcsine Law The fairer the contest, the more one side leads the whole way. A near-even split is the rarest outcome.

Bertrand's Paradox "A random chord" has three answers — 1/3, 1/2 or 1/4 — depending on how you define random.

The Sleeping Beauty Problem Halfers and thirders. Change what you count and both answers stand.

[ PERFECT FOR ]

• Math & statistics enthusiasts
• Curious minds who love "why?"
• Brain teaser & trivia fans
• Students and lifelong learners