Birthday Paradox Calculator
How many people does it take for two of them to share a birthday? Fewer than you think: with 23 people the chance is already more than 50%. Try any group size.
In a group of 23, the chance that at least two people share a birthday is 50.7%.
Assumes 365 equally likely birthdays and ignores February 29.
- Pairs of people253
- Chance someone shares your birthday5.9%
- Chance nobody shares a birthday49.3%
- Chance 3 people share a birthday1.3%
- Chance two birthdays are 1 day apart or less88.8%
- Chance two birthdays are a week apart or lessmore than 99.9%
Gives each person a random birthday and shows the dates that matched. Draw a few times to see the chance at work.
The birthday paradox explained: why 23 people are enough
The trick is that the question is about any two people, not about you. In a group of 23 there are 253 different pairs, and each pair is a fresh chance for a match. With that many chances, a shared birthday becomes more likely than not.
The chance that someone in the group shares your birthday is much lower: with 23 people it is only about 6%. You would need 254 people before that passes 50%. That gap between the two questions is what makes the result feel like a paradox.
The math assumes all 365 birthdays are equally likely and leaves out February 29; the real-life test below shows how that holds up. To see where your own birthday ranks, use How Rare Is My Birthday, and to see how many people share it, try the Birthday Twin Finder.
Worked example
Group of 23. The chance that all 23 birthdays are different is 365/365 × 364/365 × ... × 343/365, which is about 49.3%. So the chance that at least two share a birthday is 1 minus that: 50.7%.
Birthday paradox chart
The chance climbs steeply, passes 50% at 23 people and is almost certain by 70.
Birthday paradox table
| People | Pairs | Chance |
|---|---|---|
| 5 | 10 | 2.7% |
| 10 | 45 | 11.7% |
| 15 | 105 | 25.3% |
| 20 | 190 | 41.1% |
| 23 | 253 | 50.7% |
| 25 | 300 | 56.9% |
| 30 | 435 | 70.6% |
| 40 | 780 | 89.1% |
| 50 | 1,225 | 97% |
| 57 | 1,596 | 99% |
| 60 | 1,770 | 99.4% |
| 70 | 2,415 | more than 99.9% |
| 75 | 2,775 | more than 99.9% |
| 100 | 4,950 | more than 99.9% |
The birthday paradox formula
For a group of n people, the chance that nobody shares a birthday is:
P(no match) = 365/365 × 364/365 × 363/365 × ... × (365 − n + 1)/365
The chance of at least one shared birthday is then:
P(match) = 1 − P(no match)
Each new person has to miss every birthday already in the group, so each factor gets a little smaller and the product drops fast. For the chance that someone shares one particular birthday, the formula is 1 − (364/365)n − 1 instead.
Found someone who shares your birthday? What Day Was I Born shows whether you were born on the same weekday too.
A quick estimate without a calculator
Each pair of people has a 1 in 365 chance of sharing a birthday. Count the pairs, divide by 365, and use this shortcut:
P(match) ≈ 1 − e−pairs / 365
For 23 people: 253 pairs, and 253 / 365 = 0.69. e−0.69 is about 0.50, so the chance is about 50%. The exact answer is 50.7%. The shortcut is a little low for every group (for 30 people it gives 69.6% against 70.6%), but close enough to check an answer in your head.
Turned around, the shortcut gives the square root rule: a 50% chance of a match needs about 1.18 × √365 ≈ 22.5 people, so 23. The same rule works for anything picked at random. With 10,000 equally likely codes, 119 picks give an even chance that two are the same.
Three people sharing, and near misses
The same idea answers other questions. For three people on one day you need 88. For two birthdays a week apart or less, just 7.
| People | Three share | 1 day apart | 7 days apart |
|---|---|---|---|
| 7 | <0.1% | 16.1% | 60.6% |
| 10 | <0.1% | 31.5% | 87.4% |
| 14 | 0.3% | 53.7% | 98.8% |
| 23 | 1.3% | 88.8% | >99.9% |
| 30 | 2.9% | 97.8% | >99.9% |
| 50 | 12.6% | >99.9% | 100% |
| 88 | 51.1% | >99.9% | 100% |
| 100 | 64.6% | >99.9% | 100% |
| 150 | 96.5% | >99.9% | 100% |
"1 day apart" counts the same day too, and December 31 is one day from January 1. 100% means certain: 46 birthdays all 8 or more days apart would need 368 days, so from 46 people two are always a week apart or less. The calculator above shows all four chances for any group.
Does it work in real life? The World Cup test
A World Cup squad has exactly 23 players, which makes it a natural test. In 2014 the BBC checked the 32 squads against FIFA's official lists: 16 of them had at least one pair of players with the same birthday, exactly half. Adding the 2010 squads gave 31 of 64, still close to half.
Real birthdays are not spread perfectly evenly. In US data some dates have more births than others, and holidays have fewer, as the most and least common birthdays show. Uneven birthdays make a match a little more likely, not less, so the 23 still holds.
Try it with a class or a team
- Before you start, ask everyone to guess how many people it takes for a 50% chance. Most guesses are far too high.
- Go through the months: everyone born in January stands up and says their day, then February, and so on. Matches are easy to spot this way.
- Or call out birthdays one at a time and stop at the first match. On average that happens at about the 25th person (24.6).
- Leave out twins: their match is not a coincidence.
The table above covers classes of 20, 25 and 30. A larger class of 35 has a 81.4% chance, so about four classes in five will find a match.
Common mistakes
- Halving 365. 183 people sounds right for 50%, but by then a match is all but certain: with 183 people the chance is more than 99.9%.
- Expecting a match every time. 23 people give about a 50% chance, so about half the groups of 23 have no match.
- Counting people instead of pairs. Every new person can match everyone already there, so the chances add up much faster than the head count.
Where else the birthday paradox shows up
Anything picked at random from a fixed set repeats sooner than people expect. Computer security is the best known case. A short random code, or a short "fingerprint" of a file, can be matched by trying about the square root of the number of possible values, not all of them. That attack is known as a birthday attack, and it is why security codes are made long.
The same math tells you how soon two people in a group will pick the same number from 1 to 100 (13 people give an even chance).
Birthday Paradox Calculator questions
What is the birthday paradox?
The finding that 23 people are enough for a better than even chance that two of them share a birthday. In probability it is usually called the birthday problem. It is not a real contradiction, just a result that feels wrong, because most people picture someone matching their own birthday rather than any two people matching each other.
Who discovered the birthday paradox?
It is usually credited to the mathematician Harold Davenport, around 1927, but he never published it and did not believe he was the first to think of it. The first published version came from Richard von Mises in 1939. It is called a paradox only because the answer seems wrong at first, not because it contradicts itself.
How many people do you need for a 50% chance?
23 people. For a 90% chance you need 41, for 99% you need 57, and for 99.9% you need 70.
What is the birthday paradox formula?
Multiply 365/365 × 364/365 × 363/365 and so on, one factor per person, and subtract the result from 1. A small case shows how it works: for 3 people, 364/365 × 363/365 is about 99.2%, so the chance of a shared birthday is 0.8%. Each extra person adds a smaller factor, so the chance grows faster and faster.
What is the chance that someone shares my birthday?
Much lower than the chance of any match. Two people have a 1 in 365 chance (about 0.27%) of sharing a birthday. In a group of 23, the chance that one of the other 22 has your birthday is only 5.9%. For a better than even chance you need 254 people in the room, 253 besides you. The formula is 1 − (364/365) to the power of the number of other people.
How many people do you need to be sure two share a birthday?
367 people make it certain, counting February 29. With the usual 365-day assumption, 366 people are enough: there are not enough days to go around.
Does February 29 change the answer?
Hardly. Counting February 29 as a quarter as likely as other days, a group of 23 still has a better than even chance, and the chance moves by less than 0.1 percentage point.
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Published October 9, 2026. Last updated November 1, 2026. Uses the Gregorian calendar and your device's time zone.