01 · Monty Hall Problem
The Door Trap
Pick one of three doors, and the host opens a losing one. Should you switch? Your gut says 1/2; the math says switching wins 2/3 of the time. Run both strategies hundreds of times side by side.
PROBABILITY PARADOX SIMULATOR
Trust your gut,
Trust your gut,
or do the math?
Switch doors or stay? From the Monty Hall problem to the birthday paradox, Bayesian false positives and Benford's law, experience 14 famous probability paradoxes in four phases: intuition quiz, simulation, explanation and comprehension check.
Free14 paradoxes73 languagesiPhone · iPad · Android
MONTY HALL PROBLEMSimulating
1
2
3
Your first pick / opened by the host / prize
Switch66.7%
Stay33.3%
Trials —Theory 2/3 · 1/3
14probability paradoxes
4phases each
$0no in-app purchases
73languages
Paradoxes
14 probability paradoxes
3 categories: conditional probability traps, when statistics fool intuition, and expected value and decisions. 14 classic puzzles where intuition fails, tested with your own hands.
Conditional probability traps
02 · Bertrand's Box
The Gold Box
Three boxes: gold-gold, gold-silver, silver-silver. You draw a gold coin. What are the odds the other coin in that box is gold too? Not 1/2, but 2/3.
03 · Three Prisoners Problem
Warden's Verdict
One of three prisoners will be pardoned. The warden says B will be executed. Do A's chances rise to 1/2? No, they stay at 1/3, the same structure as Monty Hall.
04 · False Positive Problem
Test Trap
A 95%-accurate test comes back positive. How likely is it that you are sick? When the disease is rare, only about 16%. Count people to see Bayes' theorem at work.
05 · Sleeping Beauty Problem
Sleeping Princess
Woken once if the coin lands heads, twice if tails, with her memory erased each time. On waking, is the chance of heads 1/2 or 1/3? Explore both sides of a still-unsettled debate.
When statistics fool intuition
06 · Birthday Paradox
Miracle Classroom
How many people do you need for a better-than-even chance that two share a birthday? Most guess over 100. The answer is just 23.
07 · Simpson's Paradox
Reversed Statistics
A beats B in every group, yet B comes out ahead overall. Compare the tables and see how the reversal happens.
08 · Bus Waiting Paradox
The Endless Bus
Buses come every 10 minutes on average. Arrive at a random time and you wait 10 minutes on average, not 5. Longer gaps are easier to land in: the inspection paradox.
09 · Gambler's Fallacy
Red Chain
Red has come up again and again. Is black due? Every spin has the same odds, no matter what came before. Watch the streaks for yourself.
10 · Benford's Law
Too Many OnesNEW
In populations, areas, prices and other real-world numbers, the first digit is 1 about 30% of the time, not 1 in 9 (about 11%). See why your intuition misses.
Expected value and decisions
11 · Two Envelope Problem
Envelope Dilemma
One envelope holds twice as much as the other. Where does the "always switch" argument go wrong? Trials show that switching never changes the average.
12 · St. Petersburg Paradox
Infinite Gamble
Flip until tails, doubling the prize with every heads. The expected value is infinite, so how much would you pay to play? Watch the average refuse to settle.
13 · Newcomb's Paradox
The Predictor's Boxes
A predictor puts ¥100 million in a box only if it foresees you taking that box alone. Take both boxes, or just one? Change the predictor's accuracy and compare the payoffs.
14 · Penney's Game
The Second Mover WinsNEW
Two players each pick a pattern of three coin flips, and whichever appears first wins. Choose second, after seeing your opponent's pick, and you can win up to 7/8 of the time. A game where strength goes round in a circle, like rock-paper-scissors.
4 Phases
Four phases to real understanding
Every stage follows the same path: answer on instinct, run hundreds of trials yourself, see why your intuition was off, then finish with an applied question.
Features
Built to make it click
See it in trials
Run one trial at a time or hundreds at once. Numbers and charts update live, so the gap between intuition and result is plain to see.
Explanations
Diagrams, formulas and alternative viewpoints explain why intuition fails, and a final comprehension check confirms what you learned.
Easy to read and use
Dark and light themes. Works on iPhone, iPad and Android, and everything except ads works offline.
73 languages
English, Japanese and 71 more languages. The app follows your device language automatically.
Pricing & ads
The app is free. There are no in-app purchases, subscriptions or paid unlocks.
A Pangle banner ad is shown at the bottom of the screen, and a full-screen (interstitial) ad may be shown when you clear a stage (or leave one during the comprehension check). On iOS, the App Tracking Transparency (ATT) prompt appears after launch, and ads are personalized only if you allow tracking. On Android, the app asks once whether ads may be personalized. Either choice can be changed later from "About this app" > "Ad settings" (on iOS, this appears after you have answered the tracking prompt).