Birthday paradox calculator
Choose a group size to find the chance that any two people share a month and day.
For 23 people
At least one shared birthdayFor 23 people: classic 50.73%; historical U.S. births 50.78%.
What the two estimates mean. Classic assumes 365 equally likely dates and excludes February 29. Historical uses 85,712,738 U.S. births from 1994 to 2014 across 366 dates, including leap day. Both use independent modeled draws. The group includes everyone, and birth year does not matter. Percentages are rounded estimates, not guarantees.
23 people. 253 possible pairs.
The question is whether any pair shares a birthday
Bars: any-pair probability, starting at zero; full scale is 100%. 1 − product of distinct-date probabilities. 365 equally likely dates, independent draws, no leap day. Expected pairs is not the probability of a match.
Why do 23 people make a shared birthday likely?
In the classic model, 23 people have a 50.73% chance that at least two share a birthday. A match means the same month and day, regardless of birth year. This calculation treats birthdays as independent and gives each of 365 dates the same chance; February 29 is left out.
A room of 23 people has 253 possible pairs: 23 × 22 ÷ 2. Any one of those pairs can share a date. The pair does not have to include you.
The 253 pair events are not all independent. If A shares a birthday with B, and B shares with C, then A and C must match too. Multiplying 253 separate no-match chances would therefore give the wrong exact result. Instead, calculate the chance that all 23 birthdays are different, then subtract it from 1.
How the birthday paradox calculation works
The first person can have any birthday. For everyone to have different dates, the second has 364 possible dates out of 365. The third has 363 because two dates are already taken. The available choices shrink with each person.
For 23 people:
P(no shared birthday) = (365/365) × (364/365) × (363/365) × … × (343/365)
P(at least one shared birthday) = 1 − P(no shared birthday)
The Harvard birthday-problem solution works through this calculation and the 23-person threshold. You can repeat it for any group size. With 366 people and only 365 possible dates, at least two must match.
A worked 23-person calculation
Assume the birthdays already entered are all different. When person 23 arrives, 22 dates are occupied, leaving 343 of 365 available dates. That factor, 343/365, is the final part of the no-match product. It is not the chance that everyone in the room has different birthdays by itself.
Multiplying all the factors from the first person through person 23 gives 49.27% for no shared birthday. Subtracting that result from 100% gives 50.73% for at least one match. The checkpoints below show how the product shrinks as the group grows.
| People so far | Available dates for last person | Chance all birthdays differ | Chance at least one pair matches |
|---|---|---|---|
| 1 | 365 of 365 | 100.00% | 0.00% |
| 2 | 364 of 365 | 99.73% | 0.27% |
| 3 | 363 of 365 | 99.18% | 0.82% |
| 10 | 356 of 365 | 88.31% | 11.69% |
| 20 | 346 of 365 | 58.86% | 41.14% |
| 22 | 344 of 365 | 52.43% | 47.57% |
| 23 | 343 of 365 | 49.27% | 50.73% |
| 50 | 316 of 365 | 2.96% | 97.04% |
Read the "available dates" column assuming all earlier birthdays differ. If two earlier people already share a date, the group already has a match. Each new factor asks whether the next person avoids the distinct dates already taken.
Why expected matching pairs is not the match probability
There are 253 possible pairs in a 23-person room. Each pair has a 1/365 chance of matching in the classic model. Their expected number of matching pairs is therefore 253/365 = 0.693.
That means an average of 0.693 matching pairs across many modeled groups. It does not mean a 69.32% chance of a match. Some groups have none, while others contribute multiple pairs. Three people sharing one date, for example, contribute three matching pairs in a single group.
| People | Possible pairs | Expected matching pairs | Chance at least one match |
|---|---|---|---|
| 3 | 3 | 0.008 | 0.82% |
| 10 | 45 | 0.123 | 11.69% |
| 23 | 253 | 0.693 | 50.73% |
| 50 | 1,225 | 3.356 | 97.04% |
For larger groups the expected count can exceed one, while a probability can never exceed 100%. The expected count is useful, but it cannot replace the no-match product. The two-person explanation starts with a single pair; the three-person guide calculates the stricter event of a shared triple anywhere in a group.
Does using real birth records change the answer?
The historical U.S. distribution gives a 50.78% match chance for 23 people. The classic model gives 50.73%. They differ by about 0.05 percentage points, and both first pass 50% at 23 people.
| People in group | Classic model | Historical U.S. model |
|---|---|---|
| 10 | 11.69% | 11.71% |
| 23 | 50.73% | 50.78% |
| 30 | 70.63% | 70.68% |
| 50 | 97.04% | 97.05% |
| 100 | >99.99% | >99.99% |
For the historical model, we use each date's share of 85,712,738 recorded U.S. births from 1994 through 2014. Dates such as September 9 get their observed weight. February 29 is included at its much smaller observed share.
Our birthday distribution study measures how many dates depart from an equal daily birth reference in the same historical records. It also compares February 29 with only the years when it occurred.
The comparison changes two things at once: the date weights and whether February 29 is possible. It cannot tell us how much of the difference comes from either change. In both versions, 23 remains the first group size above even odds.
These probabilities assume independent draws from each model's birthday distribution. They are calculations, not observations of actual rooms. A family or a group of people born in the same period could have different odds.
What if I want someone to share my birthday?
If you are one of the 23 people, only the other 22 can match your birthday. A match between two other people answers a different question.
In the classic model, at least one of those 22 people matches your date with probability 5.86%. A match anywhere among the 23 has probability 50.73%.
P(someone matches your birthday) = 1 − (364/365)^22
For a date with share p, the formula becomes 1 − (1 − p)^22. September 9's historical share gives 6.45%. This is the chance under the historical model, rather than a count of living people who share that birthday.
Look up your birthday's share or see why September 9 leads the ranking. For February 29, only the historical formula applies: the classic model has no leap-day category. The leap-day guide explains its small observed share.
What are the odds of meeting someone with the same birthday and year?
The odds depend on the ages and birthdays of the people you meet. A full date-of-birth match needs the same month, day, and year. The calculator above matches month and day only.
For a person drawn from a defined group, multiply the chance of sharing your birth year by the chance of sharing your month and day among people in that year. This is the conditional-probability multiplication rule.
Both chances must describe that group. If everyone was born in your year, the year already matches. If nobody was, a full-date match is impossible. Classmates and people of many different ages need different assumptions about birth years. Those years need not be equally likely.
If each of n people independently has the same probability p of matching your full date, the chance that at least one matches is 1 − (1 − p)^n. Real encounters may not fit those assumptions. Our historical U.S. birth records do not describe everyone alive or the people you meet, so they cannot give a universal value for p.
Sources, assumptions, and downloadable results
The classic calculation follows the Harvard birthday-problem derivation. Historical date weights come from FiveThirtyEight's pinned birth files: CDC/NCHS for 1994 through 1999, then SSA for 2000 through 2014. We exclude the overlapping CDC years.
The historical calculation adds up the ways everyone could have different dates, then subtracts that probability from 1. It draws each birthday independently from the same 366-date distribution. Each weight comes from a date's share of all recorded births, not its births per calendar occurrence.
CDC/NCHS and SSA have different administrative coverage, so these historical weights do not describe everyone alive today or birthdays worldwide. Displayed results are rounded. Even 99.99% is short of certainty; a match becomes guaranteed at 366 people in the 365-date model or 367 in the 366-date model.
Download the calculation results, or read the methodology for source coverage and the birth-share definition.
Download the worked checkpoints and expected-pair calculations for the expanded formula examples.
For a known birthday, how many people share your date explains expected matches rather than the chance of any pair matching in a group.