https://mathparadoxes.com/en/ The Monty Hall problem, the birthday paradox and other puzzles where intuition fails. Play them, run simulations and follow clear explanations. en Fri, 23 Oct 2026 00:00:00 +0000 https://mathparadoxes.com/og-en.png https://mathparadoxes.com/en/ https://mathparadoxes.com/election-poll/en/ https://mathparadoxes.com/election-poll/en/ A country has 30 million voters. How many random people do you need to ask to forecast the election within ±3 points? Fri, 23 Oct 2026 00:00:00 +0000 2026-10-23T00:00:00Z statistics polls simulation A field of voter dots, a few lit up as asked, captioned n = ?. How many people does an election poll need to ask? https://mathparadoxes.com/poisoned-wine/en/ https://mathparadoxes.com/poisoned-wine/en/ One of 1,000 bottles of wine is poisoned, and the rats can test it in one round only. How many rats do you need to find the poisoned bottle for sure? Thu, 22 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z interview logic simulation A wine cellar with racks of bottles and rats beside them. One bottle in 1,000 is poisoned: how many rats do you need to find it? https://mathparadoxes.com/benford-law/en/ https://mathparadoxes.com/benford-law/en/ Take the powers of two from 2 to 2^1000. What share of them start with the digit 1? Thu, 22 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z logarithms statistics simulation The powers of two 2, 4, 8, 16, 32 and the nine possible first digits. Benford's law: what share of numbers start with 1? https://mathparadoxes.com/st-petersburg-paradox/en/ https://mathparadoxes.com/st-petersburg-paradox/en/ A coin is flipped until the first tails; the pot starts at $2 and doubles with every heads. What does the game pay on average? Thu, 22 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability expected value simulation A row of coins: heads double the pot, the first tails ends the game. The St. Petersburg paradox: what does this game pay on average? https://mathparadoxes.com/german-tank-problem/en/ https://mathparadoxes.com/german-tank-problem/en/ A factory numbers its tanks 1, 2, 3, … You come across four, numbered 4, 10, 15 and 60. How many tanks did the factory build? Thu, 22 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z statistics estimation simulation A row of tanks with factory serial numbers and four plates, 4, 10, 15 and 60, on a number line. The German tank problem: how many tanks did the factory build? https://mathparadoxes.com/inspection-paradox/en/ https://mathparadoxes.com/inspection-paradox/en/ A bus comes every 10 minutes on average, but with no timetable. You walk up to the stop at an arbitrary moment. How long do you wait on average? Thu, 22 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability statistics simulation A bus line with random gaps between buses and a rider at the stop. The bus paradox: how long is the average wait? https://mathparadoxes.com/penney-game/en/ https://mathparadoxes.com/penney-game/en/ You pick HHH, your opponent picks THH. The coin is flipped until one of the patterns comes up in a row. What are your opponent's odds? Thu, 22 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability game simulation Coin flips, heads and tails, and two patterns, HHH and THH. Penney's game: whose pattern comes up first? https://mathparadoxes.com/bertrand-paradox/en/ https://mathparadoxes.com/bertrand-paradox/en/ A chord of a circle is drawn at random. What is the chance it is longer than the side of the inscribed equilateral triangle? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability geometry simulation A circle with an inscribed triangle and a sheaf of random chords. Bertrand's paradox: how likely is a chord to beat the triangle's side? https://mathparadoxes.com/coin-streaks/en/ https://mathparadoxes.com/coin-streaks/en/ How long is the longest run of one side when you flip a fair coin 100 times? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability gambler's fallacy simulation 60 coin flips, heads and tails. How long is the longest streak in 100 flips? https://mathparadoxes.com/two-envelopes/en/ https://mathparadoxes.com/two-envelopes/en/ One envelope holds twice as much money as the other. You took one. Should you switch? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability logic simulation Two envelopes, one with twice as much money. The two envelopes paradox: switch or not? https://mathparadoxes.com/regression-to-mean/en/ https://mathparadoxes.com/regression-to-mean/en/ After praise players do worse, and after criticism they do better. Does that mean criticism works? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z statistics correlation simulation A scatter plot of scores in two rounds. Regression to the mean: does praise hurt and criticism help? https://mathparadoxes.com/coupon-collector/en/ https://mathparadoxes.com/coupon-collector/en/ An album has 50 slots, and the stickers in the packs are random. On average, how many stickers do you have to buy to fill it? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability harmonic numbers simulation A 50-sticker album with a few empty slots left. How many stickers does it take to fill it? https://mathparadoxes.com/cognitive-reflection/en/ https://mathparadoxes.com/cognitive-reflection/en/ A bat and a ball cost $1.10 together. The bat costs a dollar more than the ball. How much is the ball? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z thinking equations intuition A bat and a ball cost $1.10, and the bat costs $1 more. What does the ball cost? https://mathparadoxes.com/hundred-prisoners/en/ https://mathparadoxes.com/hundred-prisoners/en/ Each of 100 prisoners may open 50 of 100 boxes to find their number. All go free only if every one succeeds. What are the odds with the best strategy? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability permutations simulation A hundred numbered boxes. The 100 prisoners problem: each may open 50 boxes. https://mathparadoxes.com/rope-around-earth/en/ https://mathparadoxes.com/rope-around-earth/en/ A rope around the equator is made 1 metre longer and lifted evenly. How big is the gap between it and the ground? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z geometry circles intuition The Earth with a rope around it, made 1 metre longer. How high does the rope rise? https://mathparadoxes.com/friendship-paradox/en/ https://mathparadoxes.com/friendship-paradox/en/ Who has more friends: you, or your friends on average? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z networks averages simulation A friendship network with one person highlighted. The friendship paradox: who has more friends, you or your friends? https://mathparadoxes.com/buffon-needle/en/ https://mathparadoxes.com/buffon-needle/en/ A match as long as a floorboard is wide drops onto a wooden floor at random. How often does it cross a seam between boards? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z geometric probability pi Monte Carlo Matches on a striped board, the ones crossing a line highlighted. Buffon's needle: what are the odds of a crossing? https://mathparadoxes.com/two-children/en/ https://mathparadoxes.com/two-children/en/ A family has two children, and at least one of them is a boy. What is the chance both are boys? And what if at least one of them is a boy born on a Tuesday? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability conditional probability simulation Four cards of two-child families. The two-child paradox: at least one is a boy. What is the chance both are boys? https://mathparadoxes.com/simpson-paradox/en/ https://mathparadoxes.com/simpson-paradox/en/ Treatment A beats treatment B for small kidney stones and for large ones. Can B be better for all the patients together? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z statistics causality simulation Two bars of overall treatment results: treatment A 78%, treatment B 83%. Simpson's paradox: which treatment is better? https://mathparadoxes.com/secretary-problem/en/ https://mathparadoxes.com/secretary-problem/en/ 100 candidates arrive one at a time, and you must decide on each one on the spot. What chance of hiring the best? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability optimal stopping simulation A row of 20 candidates of different quality, one highlighted. The secretary problem: how to pick the best? https://mathparadoxes.com/false-positive/en/ https://mathparadoxes.com/false-positive/en/ 1% of people have a disease. The test is 99% accurate, and you test positive. What is the chance you're really ill? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability Bayes' theorem simulation A grid of 100 people, two with a positive test. The test is 99% accurate: how likely is someone really ill? https://mathparadoxes.com/birthday/en/ https://mathparadoxes.com/birthday/en/ How many people must be in a room before two of them probably share a birthday? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability combinatorics simulation A birthday calendar with two people on the same day. The birthday paradox: how many people does a match take? https://mathparadoxes.com/duckweed-lake/en/ https://mathparadoxes.com/duckweed-lake/en/ Duckweed doubles every day, and one plant covers a lake in 30 days. How long do two plants take? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z exponential growth logarithms simulation A clean lake with a few duckweed plants. One plant covers it in 30 days. What about two? https://mathparadoxes.com/monty-hall/en/ https://mathparadoxes.com/monty-hall/en/ You're on a game show and pick one of three doors. The host, who knows where the car is, opens another door to reveal a goat and offers a switch. Should you take it? Mon, 19 Oct 2026 00:00:00 +0000 2026-10-22T00:00:00Z probability game simulation Three doors, the first one picked, a goat behind the third. The Monty Hall problem: should you switch?