By the Age-Finder team · Last updated 7 November 2026
Quick answer
With 23 people in a room, the chance that two of them share a birthday is already over 50%. With 70 people it is 99.9%.
Type any group size from 2 to 999 into the calculator to see the exact chance, the real-world chance from U.S. birth records, and how many people it takes to reach 50% and 99%.
How to Use the Birthday Paradox Calculator
Type the number of people in the group, from 2 to 999, and select Calculate. The result shows the chance of a shared birthday, the same chance using real birth data, and how many people you need for a 50% and a 99% chance.
Four quick chips sit under the form: 23 people, A class of 30, 50 people and 70 people. One tap fills the group size in and calculates at once. If you type a number outside 2 to 999, the calculator answers: “Enter a group size from 2 to 999 people.”
The big number is the classic chance, worked out as if every birthday were equally likely. The real-world chance sits beside it, and the note under the result says that real birthdays are not spread evenly (September is busiest), so the real-world chance is slightly higher.
Above 365 people the result says a shared birthday is certain, because there are more people than days in the year.
Why It Feels Like a Paradox
Most people guess that you would need around 183 people, half of 365, for an even chance of a shared birthday. The real number is 23. The surprise comes from the number of pairs: 23 people can form 253 different pairs, and each pair is a chance for a match.
The question is not “does someone share my birthday?”, which does need a large group. It is “do any two people in the group share a birthday?”, and that grows much faster.
It is called a paradox, but it is not a contradiction. Mathematicians call it a veridical paradox: a result that seems wrong at first glance and is true. The maths is airtight; it is human intuition that fails. The problem is usually credited to the Austrian mathematician Richard von Mises, who discussed it in 1939.
How the Chance Is Calculated
The classic method assumes 365 equally likely birthdays and ignores 29 February. It is easiest to work out the chance that everyone has a different birthday, then subtract it from 1. The first person can have any birthday: 365 out of 365.
The second must avoid one day: 364 out of 365. The third must avoid two days: 363 out of 365, and so on. Multiply these together for everyone in the group, and subtract the result from 1.
Written as a formula, with n people: P(shared) = 1 – 365! / ((365 – n)! x 365^n). For 23 people the product is (365 x 364 x 363 x … x 343) / 365^23 = 0.4927, so the chance of a shared birthday is 1 – 0.4927 = 0.5073, or 50.7%.
| People | Pairs | Classic chance | Real-world chance |
|---|---|---|---|
| 2 | 1 | 0.3% | 0.3% |
| 5 | 10 | 2.7% | 2.7% |
| 10 | 45 | 11.7% | 11.7% |
| 15 | 105 | 25.3% | 25.3% |
| 20 | 190 | 41.1% | 41.2% |
| 22 | 231 | 47.6% | 47.6% |
| 23 | 253 | 50.7% | 50.8% |
| 30 | 435 | 70.6% | 70.7% |
| 40 | 780 | 89.1% | 89.2% |
| 50 | 1,225 | 97.0% | 97.1% |
| 57 | 1,596 | 99.0% | 99.0% |
| 70 | 2,415 | 99.9% | 99.9% |
| 100 | 4,950 | more than 99.999% | more than 99.999% |
The milestones to remember are 23 people for 50%, 41 for 90%, 57 for 99%, 70 for 99.9% and 80 for 99.99%. The calculator shows the 50% and 99% figures for every group size you enter. Above 99% it shows three decimals, so 70 people reads 99.916% and 57 reads 99.012%.
The Real-World Version
Real birthdays are not spread evenly. In U.S. records from 1994 to 2014, mid-September dates are the most common and holidays such as 25 December the least.
The most common birthday of all is 9 September, with 12,344 births a day on average. The least common is 25 December, with 6,601. 29 February averages 10,512, counted a quarter as often because it only exists in leap years.
When some dates are more popular, matches become a little more likely. The calculator works out this real-world chance exactly from the average births on each date, including 29 February.
The difference is small: for 23 people, the real-world chance is 50.8% instead of 50.7%. The gap shrinks as the group grows: at 70 people it is 99.9167% against 99.9160%.
There is a useful rule hiding here. The uniform distribution is the hardest case for a match: any unevenness increases the chance, so 23 is a slight overestimate for real people. The classic answer is the lowest the real answer can be.
The Quick Approximation
For a fast estimate without multiplying 23 fractions, use P is about 1 – e^(-n^2 / 730). For 23 people: 1 – e^(-529 / 730) = 1 – e^(-0.725) = 0.5155, about 51.6%, close to the exact 50.7%.
For 10 people it gives 12.8% against the exact 11.7%, and for 50 it gives 96.7% against 97.0%. Good enough for a quick guess, not a substitute for the exact figure.
Solve it backwards for the group size: n is about 1.18 x the square root of the number of days. For 365 days: 1.18 x 19.1 = 22.5, so about 23 people.
The same rule works for any number of days: with 1,000 possible days it takes about 37 people (the exact chance at 37 is 49.1%), and with 10,000 possible days it takes about 118 (the exact chance at 118 is 50.0%). With 366 days, leap years included, 23 people still pass 50%, at 50.6%.
Shared Birthdays Versus Your Birthday
Two different questions are easy to mix up. The chance that any two people in a group share a birthday, which this calculator answers, passes 50% at 23 people. The chance that someone in a group shares your own birthday needs 253 other people to pass 50%, because only pairs that include you count.
The first question counts every possible pair; the second only the pairs that include you. With 23 people there are 253 pairs in total but only 22 that include you, which explains the gap.
| Other people | Chance someone shares your birthday |
|---|---|
| 22 | 5.9% |
| 100 | 24.0% |
| 253 | 50.1% |
| 365 | 63.3% |
| 500 | 74.6% |
| 1,000 | 93.6% |
The formula for this question is simpler: with n other people, the chance is 1 – (364/365)^n. For 22 other people that is 1 – (364/365)^22 = 5.9%. Even with 1,000 other people, about 1 in 16 birthdays has nobody sharing it.
Try It in Real Life
Classrooms, sports teams and offices make good tests. A class of 30 pupils has about a 70% chance of a shared birthday.
A football (soccer) match has 22 players and a referee, 23 people in all, just over an even chance. A group of 50 colleagues is almost certain to include a shared birthday, about 97%.
The 2014 FIFA World Cup ran the experiment for real. Each of the 32 squads had exactly 23 players, and 16 squads had at least one pair sharing a birthday, exactly half. Five of those had two pairs: Argentina, France, Iran, South Korea and Switzerland.
The other eleven with one pair were Australia, Bosnia and Herzegovina, Brazil, Cameroon, Colombia, Honduras, Netherlands, Nigeria, Russia, Spain and USA. The theory predicts 50.7% of 32, about 16.2 squads, and the count was 16.
Including the 2010 squads, 31 of 64 squads had shared birthdays. Researchers at the University of Tubingen had already tested the 2006 World Cup and found that in 53% of games, at least two players on the field shared a birthday.
Two more numbers from the same maths. If you add people to a room one at a time, the average number you need before the first match appears is 24.6, while the median is 23.
And if you want to collect every one of the 365 birthdays, the coupon collector’s answer is about 2,365 people: 365 times the sum of 1 + 1/2 + 1/3 + … + 1/365.
Where the Idea Is Used
The same mathematics appears outside birthdays. In computer security, the “birthday attack” uses it to estimate how quickly two inputs can produce the same hash value. In data processing, it helps estimate the chance of duplicate random IDs. In statistics teaching, it is a classic example of how intuition underestimates probabilities.
The hash version is the one that matters most. For a hash with n bits of output, a collision becomes likely after about 2^(n/2) tries, not 2^n, because the inputs pair up the same way people do.
A 256-bit hash only gives 128 bits of collision resistance, which is why hashes are built twice as wide as the keys they protect.
| Hash | Output size | Tries for a 50% collision |
|---|---|---|
| CRC-32 | 32 bits | about 65,000 (2^16) |
| 64-bit hash | 64 bits | about 4.3 billion (2^32) |
| MD5 | 128 bits | about 2^64 (18.4 quintillion) |
| SHA-1 | 160 bits | about 2^80 |
| SHA-256 | 256 bits | about 2^128 |
MD5 fell much further than brute force: researchers published practical collisions in 2004, in about an hour of computation, running at 2^39 instead of the expected 2^64. SHA-1 followed in 2017, when the SHAttered project published two different PDF files with the same hash. SHA-256 has no practical collision attack known today.
Why the Numbers Grow So Fast
The number of pairs grows roughly with the square of the group size: 10 people make 45 pairs, 23 people make 253 and 50 people make 1,225. Each pair has about a 1 in 365 chance of matching, so once a group has a few hundred pairs, at least one match becomes likely.
That is why the chance climbs from 11.7% for 10 people to 97% for 50. Doubling the people roughly quadruples the pairs, which is why the curve is steepest between 20 and 50 people and almost flat after 70.
How This Calculator Compares
The three calculators below rank for the same query, and each one draws something this page does not: a curve, a slider, or a simulator. The table shows what each one offers that this one leaves out.
| Feature | Age-Finder | Unspurious | The Probability Lab | Math.tools |
|---|---|---|---|---|
| Free, no sign-up, nothing to install | Yes | Yes | Yes | Yes |
| Group sizes above 365 | Yes, to 999 | Yes | To 366 | Not shown |
| Real-world chance from actual birth records | Yes | No | No | No |
| Probability curve | No | Yes | Yes | Not shown |
| Reverse mode (target chance to group size) | No | No | No | No |
| Day-count option (365 or 366 days) | No | Yes | No | No |
| Chance someone shares your birthday | No | Yes | Not shown | Not shown |
| Random-room simulator | No | No | No | Yes |
The gaps in the tool, stated plainly: no probability curve, no reverse mode, no day-count option, no answer to the “shares your birthday” question, and no simulator.
The guide covers all five by hand where it can: the probability table gives the curve in numbers, the approximation rule lets you work backwards to a group size,
The 366-day figure is given beside the 365-day one, the “your birthday” table runs to 1,000 people, and the World Cup squads are 32 real rooms of 23. A curve, a reverse mode and a simulator wait on the tool itself.
Frequently Asked Questions
How many people do you need for a 50% chance of a shared birthday?
23 people, assuming all 365 birthdays are equally likely. With real birth data the answer is still 23, at 50.8% instead of 50.7%.
How many people for a 99% chance?
57 people give about a 99% chance, 70 people about 99.9%, and 80 people about 99.99%. At 100 people the chance is more than 99.999%.
Is this the chance someone shares my birthday?
No. That is a different question. For a group of n other people, the chance that at least one shares your birthday is 1 minus (364/365) to the power n, so you need 253 other people for an even chance. With 22 other people the chance is only 5.9%.
Does 29 February change the answer?
Very slightly. The classic answer ignores it. The real-world figure includes it, weighted by how rarely it occurs, at a quarter of a normal day. With 366 days counted, 23 people still pass 50%, at 50.6%.
Can the chance ever be 100%?
Yes. With 366 people, two must share a birthday, because there are only 365 possible dates. The classic method, which uses 365 days, reaches certainty at 366 people. Counting 29 February as a 366th date, certainty comes at 367 people.
Why is the real-world chance higher?
When some birthdays are more common than others, people are more likely to land on the same popular dates, so matches happen slightly more often than if every date were equally likely. The most common birthday is 9 September and the least common is 25 December, so September birthdays collide more often than December ones.
How do I explain the paradox to a class?
Ask everyone to say their birthday in turn. With 23 or more pupils, there is a better than even chance of a match, and with 30 about a 70% chance, which usually surprises the room. Then count the pairs on the board: 23 pupils make 253 pairs, and that number is the whole explanation.
What is the birthday paradox formula?
With n people and 365 equally likely birthdays: P(shared) = 1 – 365! / ((365 – n)! x 365^n). The quick approximation is P is about 1 – e^(-n^2 / 730), and the group size for a 50% chance is about 1.18 x the square root of the number of days.
Why is it called a paradox if it is not a paradox?
It is a veridical paradox: a result that seems wrong at first glance and is true. There is no contradiction in the maths, only a gap between what intuition expects and what the numbers say. The problem is usually credited to the Austrian mathematician Richard von Mises, who discussed it in 1939.
Keep Exploring
How Common Is My Birthday?: see how common your own birthday is, ranked against all 366 dates from the same U.S. birth records.
Average Age Calculator: work out the mean age of a group, with the median and the range worked by hand.
Date of Birth Calculator: find a date of birth from an age, or an age from a date of birth.
Life in Numbers: see your life counted out in days, weeks, heartbeats and breaths.
Sources
FiveThirtyEight: U.S. births dataset (CDC NCHS and SSA) (the birth data used for the real-world chance)
Encyclopaedia Britannica: Statistical independence (the birthday problem and the von Mises 1939 attribution)
Math.tools: The Birthday Paradox: Why 23 People Are Enough (the approximation formula and the square-root rule)