Math Paradoxes, Interactively: Probability Puzzles That Fool Intuition

Probability · 3 doors

The Monty Hall problem

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?

Intuition50 / 50

Actually2/3 to switch

The most famous puzzle about conditional probability. In 1990 thousands of readers, including mathematicians with PhDs, insisted the correct answer was wrong.

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Exponential growth · 30 days

The duckweed lake puzzle

Duckweed doubles every day, and one plant covers a lake in 30 days. How long do two plants take?

Intuition15 days

Actually29 days

The classic lily pad problem about exponential growth and doubling, and why intuition gets it wrong.

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Combinatorics · 365 days

The birthday paradox

How many people must be in a room before two of them probably share a birthday?

Intuition≈ 183 people

Actually23 people

A puzzle about coincidences where intuition is off many times over. The same idea powers the "birthday attack" on cryptographic hashes.

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Bayes' theorem · medical tests

The false positive paradox

1% of people have a disease. The test is 99% accurate, and you test positive. What is the chance you're really ill?

Intuition99%

Actually50%

A puzzle about the base rate fallacy: what a positive result from an accurate test really means when a disease is rare.

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Optimal stopping · 100 candidates

The secretary problem: when to stop

100 candidates arrive one at a time, and you must decide on each one on the spot. What chance of hiring the best?

Intuition≈ 1%

Actually37%

The classic optimal stopping problem: how many candidates to let go before you choose, and what chance that gives you.

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Statistics · a 2×2 table

Simpson's paradox

Treatment A beats treatment B for small kidney stones and for large ones. Can B be better for all the patients together?

IntuitionNo, A wins

ActuallyB: 83% vs 78%

A classic statistics paradox on real kidney stone data: how to read overall percentages, and why decisions need a causal picture.

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Conditional probability · 2 children

The two-child paradox

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?

Intuition1/2

Actually1/3 and 13/27

Martin Gardner's classic conditional probability puzzle: what "at least one is a boy" really tells you, and whether the weekday of birth matters.

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Geometric probability · π

Buffon's needle

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?

Intuition≈ 50%

Actually2/π ≈ 63.7%

A random experiment with no circle in sight that produces the number π. One of the earliest examples of the Monte Carlo method.

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Networks · averages

The friendship paradox

Who has more friends: you, or your friends on average?

Intuitionabout the same

Actuallyyour friends

Scott Feld's network paradox: how to compare your friend count with your friends' fairly, the formula E[d²]/E[d], and the same effect in classes, buses and epidemics.

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Geometry · 40,075 km

The rope around the Earth

A rope around the equator is made 1 metre longer and lifted evenly. How big is the gap between it and the ground?

Intuitionmicroscopic

Actually15.9 cm

A puzzle about circumference: how far one extra metre of rope lifts it above a sphere, from an orange to the Sun.

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Permutations · 100 boxes

The 100 prisoners problem

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?

Intuition≈ 0%

Actually31.2%

A puzzle about random permutations and cycles: can a strategy save prisoners who cannot confer during the search?

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Psychology of thinking · 3 questions

The bat and ball: the cognitive reflection test

A bat and a ball cost $1.10 together. The bat costs a dollar more than the ball. How much is the ball?

Intuition10 cents

Actually5 cents

The three questions of Shane Frederick's test, the kind you want to answer instantly. About System 1 and System 2 and the habit of checking your first thought.

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Probability · 50 stickers

How many stickers to fill the album

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?

Intuition≈ 70 stickers

Actually≈ 225 stickers

The coupon collector's problem: how many packs a full album takes, and how swapping duplicates helps.

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Statistics · skill + luck

Regression to the mean

After praise players do worse, and after criticism they do better. Does that mean criticism works?

IntuitionYes, it works

ActuallyIt would happen anyway

A puzzle about skill, luck and what really changes the result after an unusually good or bad game, and why medical trials need a control group.

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Probability · reasoning

The two envelopes paradox

One envelope holds twice as much money as the other. You took one. Should you switch?

Intuitionswitch: +25%

Actuallyno difference

The sum "the other holds 2A or A/2, so 1.25A" looks airtight. We test whether it holds and what changes once you see the amount inside.

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Probability · 100 flips

Coin streaks

How long is the longest run of one side when you flip a fair coin 100 times?

Intuition3–4 flips

Actually≈ 7 flips

What randomness really looks like: runs of one side, the gambler's fallacy and the hot hand in basketball.

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Geometric probability · chords

Bertrand's paradox

A chord of a circle is drawn at random. What is the chance it is longer than the side of the inscribed equilateral triangle?

Intuitionone answer

Actually1/3 · 1/2 · 1/4

Joseph Bertrand's 1889 puzzle about geometric probability and what "at random" really means.

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Probability · a coin game

Penney's game

You pick HHH, then your opponent picks THH. The coin is flipped until one of the patterns comes up in a row. What are your opponent's odds?

Intuition50/50

Actually7/8

A coin game won by whoever chooses second: every pattern has a stronger rival, as in rock, paper, scissors.

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Probability · waiting

The bus paradox

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?

Intuition5 minutes

Actually10 minutes

The inspection paradox: why you land in long gaps more often, why transit needs timetables, and why students see bigger classes than the dean does.

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Statistics · 4 serial numbers

The German tank problem

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?

Intuition60

Actually≈ 74 tanks

How a few serial numbers reveal total output, and why this count beat Allied intelligence.

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Probability · a coin and a pot

The St. Petersburg paradox

A coin is flipped until the first tails. The pot starts at $2 and doubles with every heads, and at the end you take it. What does the game pay on average?

Intuitiona few dollars

Actuallyinfinity

A game with an infinite average win that nobody would pay $100 for. How Bernoulli, Cramer and Feller explained it.

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Statistics · first digits

Benford's law

Take the powers of two 2, 4, 8, 16, … up to 2¹⁰⁰⁰. What share of them start with the digit 1?

Intuition1/9 ≈ 11%

Actually30.1%

Why 1 comes first three times more often than it seems: the log scale, compound interest and products of random numbers.

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Logic · 1,000 bottles

The poisoned wine and the rats

One of 1,000 bottles of wine is poisoned. The poison kills a rat in about 20 hours, and the feast is in a day, so there is time for one test only. Each rat can taste as many bottles as you like. How many rats do you need to find the poisoned one for sure?

Intuition999

Actually10

An interview puzzle with group testing behind it: from blood tests to hunting bugs in code.

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Statistics · 30M voters

How many people should a poll ask

A country has 30 million voters. How many random people do you need to ask to forecast the result within ±3 points in 95 polls out of 100?

Intuitionhundreds of thousands

Actually1,068

Why the sample size does not depend on the size of the country, and why millions of answers from "convenient" people miss by more than a thousand random ones.

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Logic · 100 floors

Two eggs and 100 floors

A 100-floor building and two identical eggs. From some floor up an egg breaks, below it stays whole, and a whole egg can be dropped again. How many drops do you need in the worst case to find the highest safe floor for sure?

Intuition50

Actually14

An interview puzzle about a search where every failure has a price: from load tests to crash tests.

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Logic · 12 coins

Twelve coins and a balance

Among 12 identical-looking coins one is fake: lighter or heavier, nobody knows. You have a two-pan balance and no weights. How many weighings do you need to find the fake for sure and say whether it is lighter or heavier?

Intuition4–6

Actually3

An interview puzzle with base 3 behind it, and the rule that builds decision trees in machine learning.

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Logic · 100 hats

Hats in a line

100 people stand in a line, each in a black or white hat. Everyone sees the hats in front, but not their own. Starting from the back, each says one colour aloud. They may agree on a strategy in advance. How many can surely name the colour of their own hat?

Intuition50

Actually99

An interview puzzle with the parity bit, RAID arrays and a bank card's check digit behind it.

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Logic · 5 pirates

Pirates and 100 gold coins

Five pirates of different rank split 100 coins. The most senior proposes a split, and everyone votes. If at least half vote yes, that is the split; if not, he goes overboard and the next one proposes. The pirates are perfectly logical and greedy. How many coins does the captain keep?

Intuition20

Actually98

An interview puzzle about backward induction: reasoning from the end, which game theory is built on.

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Logic · 100 + 100 islanders

The blue-eyed islanders

An island has 100 people with blue eyes and 100 with brown. There are no mirrors, nobody talks about eye colour, and anyone who works out their colour must leave at midnight. All are perfect logicians. A visitor says in front of everyone: "I can see someone with blue eyes." Everyone already knew that. What happens?

Intuitionnothing

Actuallynight 100

An interview puzzle about common knowledge: from muddy children to the two generals and TCP.

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Logic · 25 horses

25 horses and 5 lanes

There are 25 horses and a track for only 5. There is no stopwatch: a race shows only the finishing order. Each horse always runs the same, and all speeds differ. How many races do you need to find the three fastest and their order for sure?

Intuition6

Actually7

An interview puzzle about what exactly each race proves. Behind it stand Lewis Carroll's tennis tournaments and selection algorithms.

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Logic · 4 people, 1 torch

The bridge and the torch

At night four people must cross a narrow bridge that holds at most two. There is one torch, and nobody can walk the bridge without it. They cross in 1, 2, 5 and 10 minutes, and a pair walks at the slower one's pace. What is the fewest minutes for all of them to reach the other side?

Intuition19

Actually17

An interview puzzle about how the best plan is not made of the most convenient steps.

Cross the bridge →

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Game theory · 4,000 drivers

Braess's paradox

4,000 drivers use two routes, 2,000 on each, and the trip takes 65 minutes. The city opens a free shortcut between the routes that takes no time at all. Each driver picks the fastest way for themselves. How long will the trip take now?

Intuition, min< 65

Actually, min80

A game theory puzzle about how a new road can slow everyone down and why cities sometimes close streets.

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Statistics · 1943

Survivorship bias: Wald's planes

Bombers come back with holes in the fuselage and wings, and hardly any on the engines. There is armour for two parts of the plane. Where should it go?

Intuitionfuselage and wings

Actuallyengines and cockpit

Why the holes on returning planes show where a hit is safe, and how the same mistake flatters mutual funds and billionaires' advice.

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Conditional probability · a live debate

The Sleeping Beauty problem

On heads Beauty is woken once, on tails twice, and her memory is wiped each time. She has just woken. What is the chance of heads?

Intuition1/2, it's a fair coin

Actually1/3 per awakening

Half the experiments end in heads, yet only a third of the awakenings follow heads. We look at which question each of the two camps of philosophers is answering and whose bets pay off.

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Probability · roulette and a random walk

Gambler's ruin

You have $100. You bet $1 at a time on red at roulette until you have $200 or run out of money. Red comes up in 18 spins out of 37. What is the chance of doubling your money?

Intuitionalmost 50%

Actually0.45%, 1 in 224

Each bet is almost fair, but there are thousands of them, and the house edge eats almost everything. Why a single $100 bet gives 48.6%, and why boldness beats caution here.

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Statistics · selection

Berkson's paradox

In town, looks and kindness are unrelated. You only date people whose looks and kindness together clear a bar. Are the two linked among your dates?

IntuitionNo, there is no link

ActuallyBetter-looking, less kind

A puzzle about how selection alone creates a link between independent traits, and why hospital data, university admissions and hiring can show patterns that do not exist in real life.

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Statistics · A/B testing

The A/A test: two identical pages

The same page is shown as "A" and as "B", converting at 5%, with 1,000 visitors each. How often will one of them "beat" the other by 20% or more?

Intuitionrarely, a few %

Actually35.8%

Why 1,000 visitors means only 50 purchases, how noise paints "winners", and how much traffic you really need to spot +10%.

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Statistics · A/B tests

Peeking at an A/B test

Two identical pages, a 30-day test. The marketer checks the dashboard every evening and stops the test at the first "95% confidence". How often is one of the identical pages declared the winner?

Intuition5%

Actually28%

Why every look at a test gives the noise one more chance, and what to do if you can't wait for the full sample.

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Statistics · A/B tests

Slice it until something wins

New email B is no better than the old A. The analyst checks the result separately in 20 countries at p < 0.05. What is the chance that at least one country shows a "significant" difference?

Intuition5%

Actually64%

Why every new slice, metric and variant buys the noise another lottery ticket, and how to keep a false insight out of your report.

Hunt for a winner yourself →

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Dmytro Kudrenko

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