Five questions · ten thousand trials each
The Odds
Your intuition about chance is wrong in specific, repeatable ways. Commit to an answer, and then watch a simulation run thousands of trials in front of you and settle on the truth right next to your guess.
The rounds
Round 1 of 5
Running
All five rounds
About This Game
The Odds is five questions about chance, each one chosen because the correct answer offends most people the first time they hear it. You are made to commit before you see anything, which matters: a probability puzzle you have already been told the answer to teaches you nothing, and half the reason these are famous is how strongly people defend the wrong number.
Then, instead of proving it with algebra, the page brute-forces it. Thousands of rooms are filled with randomly-birthdayed strangers. Thousands of game show doors are shuffled and opened. Two hundred coins are flipped ten thousand times over. The running estimate is drawn as a chalk line that lurches about wildly at first and then, as the trials pile up, stops arguing and lies down on the answer.
That convergence is the actual subject. It is one thing to be told that switching doors wins two times in three; it is another to watch a line stagger around 40 per cent, drift, and then settle onto 66.7 and refuse to leave. Probability stops being a claim someone is making at you and becomes something you saw happen.
How To Play
- Read the question. Do not look it up — a guess you had to make is worth more than an answer you borrowed.
- Drag the slider and lock it in. There is no undo, deliberately.
- Watch the simulation run. The wobble at the start is the honest part; small samples really are that unreliable.
- Read the reveal, then move on. Five rounds takes about four minutes.
- At the end you get all five side by side. The pattern of which ones you missed is more interesting than how many.
Why This Exists
Because there is a specific shape to how people get chance wrong, and it is worth seeing your own version of it. Most of us are decent at the middle of a distribution and hopeless at its edges. We badly underestimate how quickly coincidences become likely when there are many chances for them — that is the birthday question. We badly underestimate how much structure pure randomness produces — that is the run of seven heads. And we are catastrophic at rare events, which is the one that matters outside a game.
That fourth round, where a test that is right almost every time still gets it wrong fifty times out of fifty-one, is not a trick. It is the single most consequential piece of arithmetic most people never do. Screening, security alerts, fraud flags, any search for something rare: the accuracy of the test is almost beside the point next to the rarity of the thing. People's lives turn on that number and the intuition it contradicts is nearly universal.
There is a good precedent for needing to see it rather than be told. When the door problem ran in a magazine column in 1990, more than a thousand people with doctorates wrote in to say the correct answer was wrong. Paul Erdős, one of the most formidable mathematicians of the twentieth century, would not accept it either — not until somebody sat him down in front of a computer simulation. Which is, more or less, what this page is.
Frequently Asked Questions
What is The Odds?
Five probability questions where almost everybody's intuition is wrong. You commit to an answer first, and only then does a Monte Carlo simulation run thousands of trials on screen, drawing its running estimate as a chalk line that wanders and then settles onto the truth next to your guess.
Is the simulation real or just an animation?
Real, and running in your browser as you watch. Every trial is an actual random draw: rooms of people given random birthdays, doors shuffled and opened, two hundred coins flipped, athletes tested. The line is paced so it takes a few seconds rather than a few milliseconds, because the whole point is watching a noisy estimate become a confident one.
Why is the answer to the birthday question only 23?
Because you are not comparing yourself against everyone else, you are comparing every pair. Twenty-three people make 253 different pairs, and each pair gets its own chance to match. At 23 the probability crosses just past even at 50.7 per cent, and by 50 people it is 97 per cent.
Why does switching doors win two times in three?
Your first pick is right one time in three, and nothing the host does afterwards changes that. What his move changes is the other two doors: between them they hold two chances in three, and by opening the one he knows is empty he pours all of it onto the single door left. Switching is a bet that your first guess was wrong, which it usually was.
Do I need to know any maths?
None at all. Every answer is arrived at by brute force rather than algebra, which is exactly the point: you can watch the truth emerge from repetition without following a single line of proof. The exact figures are given afterwards for anyone who wants them.
Does it work on mobile?
Yes. Every answer is given with a single slider, so the whole game is playable with one thumb, and the chalk curves are drawn to fit a phone screen. Nothing is saved, uploaded or scored against anyone else.
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