What is the birthday paradox?
The birthday paradox is the classic probability puzzle of how many people it takes before two of them probably share a birthday. It earns the name because the exact answer clashes with almost everyone’s intuition. This test lets you find out how good your own gut is at it: you guess first, with no hints, and only then see the exact answer and how many times off you were. Teachers and psychologists have put the same question to students and casino staff for decades, and the result sheet shows how your guesses compare.
There are four questions, each answered with a head-count: how many people for a 50% chance that two share a birthday, how many for a 99% chance, how many others for a 50% chance that someone shares your birthday, and how many for a 50% chance that three share one. You swipe passengers on and off a commuter carriage, up to 400 of them. After each answer the page simulates 1,000 carriages at your number and 1,000 at the true one, lights a lamp for every carriage with a match, and gives the exact probability alongside. Each answer is scored by how many times off it is, too many or too few, and your score is the geometric mean of the four.
It is not a maths exam or an IQ test. It measures a gut feeling for chance, and most people miss these questions by a mile. Like the classic problem, it assumes a 365-day year with every birthday equally likely and leaves out 29 February. Real birthdays are a little uneven through the year, and any unevenness only makes a match more likely, so real rooms match slightly sooner than these answers.
What the test asks
One thing, asked four ways: how good your gut is at guessing when birthdays collide.
The task
- Four questions from the classic birthday problem: two people sharing a birthday at 50% and at 99%, someone sharing your birthday at 50%, and three people sharing one at 50%
- Answer with a head-count: swipe up or down on the carriage, scroll, or use the − / + bar (tap for one person, hold to speed up, drag the bar, arrow keys, Shift for ten at a time)
- After each answer, 1,000 random carriages run at your number and at the true number, one lamp per carriage, lit where the match happens; then a real matching pair is linked inside the carriage
- One answer per question, no retries
Your result
- How many times off you were on average, and how many of the four answers were close (within 1.25×)
- A tier from “Right on time” (within 1.1×) to “Wrong train entirely”, and whether you mostly guessed too many or too few
- A ticket with each of your answers, the true answer and the exact chance at your number, plus curves of how the chance climbs with the head-count
- Why the true answers land where they do, and how your first answer compares with what people usually guess, from Lesser (1999) and Voracek, Tran & Formann (2008)
- A shareable card with your score, and a challenge link: a friend answers the same four questions and you see who got closer
Before you start
- About two minutes, four questions
- Works on a phone (swipe) or a computer (drag, scroll or arrow keys); it needs a browser with 3D graphics (WebGL) switched on
- Sound effects are optional and can be switched off
- Free, with no sign-up
Birthday paradox FAQ
Is this test free? Do I need to sign up to see my result?
Yes, it’s free, and there’s nothing to sign up for: no account, no email, no payment. Your result appears on this page the moment you finish, and you can share it or play again straight away.
How many people do you need for two to share a birthday?
That is the first question in this test, so the page lets you guess before it tells you. Board as many passengers as you think it takes for a 50% chance that two share a birthday, and the exact number appears straight after, with 1,000 simulated carriages run at your guess and 1,000 at the true answer. The calculation assumes a 365-day year with every birthday equally likely.
Why is it called the birthday paradox?
The birthday paradox is not a contradiction in the maths: the answer is exact and easy to check. It is called a paradox because the true head-count feels wrong to almost everyone who hears it. The test explains the reason on the result sheet, after you have committed to all four guesses, because knowing it first would change what you guess.
What will the birthday paradox test tell me?
The birthday paradox test tells you how good your intuition for coincidence is: how many times off each of your four head-counts was, whether you leaned one way, and a tier from “Right on time” to “Wrong train entirely”. The result sheet also compares your first answer with what people typically guess, from Lesser’s 1999 classroom report and a 2008 study of 721 psychology students and 73 casino visitors and staff by Voracek, Tran and Formann.
How is the birthday paradox test scored?
The birthday paradox test scores each answer by how many times off it is, too many or too few, and averages the four with a geometric mean. Within 1.1× counts as spot on and within 1.25× as close. The chance shown for your number is calculated exactly; the 1,000 carriages are a live simulation, so their lamp count wobbles a little.
Does the birthday paradox hold with real birthdays?
Yes, and real rooms match slightly sooner. The classic problem, like this test, uses a 365-day year with every date equally likely and no 29 February. Real births are a little uneven through the year, and any unevenness only makes a shared birthday more likely.
Is the birthday paradox test a maths or IQ test?
No. The test measures a gut feeling for chance, not intelligence or maths ability, and most people miss these questions by a mile. One run is four guesses, so the score is a snapshot of your intuition, not a grade.
How long does the birthday test take, and does it work on a phone?
About two minutes for four questions, on a phone or a computer. On a phone you swipe people on and off the carriage; on a computer you drag, scroll or use the arrow keys. The test needs a browser with 3D graphics (WebGL) switched on. A challenge link carries your result and the name you type in the part of the address after #.
Where the numbers come from
The true answers are exact calculations for a 365-day year with equally likely birthdays, not survey results. What people usually guess comes from the two sources below, read as abstracts; neither gives a typical guess for every question, so the test shows no percentiles. The site does not publish its own players’ data.
How it’s scored
Exact answers
The true answers are calculated, not surveyed. Two sharing: 1 minus the chance that everyone’s birthday differs. Someone sharing yours: 1 − (364/365)ⁿ, since each stranger misses your day with chance 364/365. Three sharing: count every way to hand out n birthdays with no day used three times, and subtract from 1.
Times-off factor
Each answer is scored by how many times too high or too low it is, whichever way: guessing 120 when the answer is 60 and guessing 30 are both 2× off. Within 1.1× counts as spot on, within 1.25× as close.
Geometric mean
Your overall score multiplies the four factors together and takes the fourth root, so twice too many and half too few weigh the same.
The 1,000 carriages
A real random simulation every time, with each passenger’s birthday drawn at random. The number of lit lamps wobbles from run to run; the probability in the table is the exact one.