How to use it
- Enter the Number of people in the group, for example 23.
- Keep Days in the year at 365 unless you want to count 29 February as a birthday.
- Type a Target chance, such as 50, to find the group size that reaches it.
- Read Chance that two share a birthday and the chart to see how fast the chance grows.
How the chance is calculated
Add the people one at a time. The first person can have any birthday. The second must avoid that day, which is a chance of 364 in 365. The third must avoid two days, and so on. Multiplying these chances gives the chance that everyone differs, and taking that from one gives the chance of a shared birthday.
P(shared) = 1 − (365/365 × 364/365 × … × (365 − n + 1)/365), for n people.- The product is computed exactly with whole numbers, so nothing is rounded on the way.
- With more people than days, the chance is certain, because two people must then share a day. That is the pigeonhole principle.
- The target search starts with one person and stops at the first group size whose chance reaches the target.
What the model leaves out
- Twins and multiple births, which make a shared birthday more likely than the model says.
- Seasonal and weekly patterns in births. Real birthdays are not spread perfectly evenly, and an uneven spread raises the chance a little.
- Any meaning of the match. The chance counts any shared day, not a shared birthday of a particular person.
Worked example: 23 people
With 23 people there are 23 factors, from 365/365 and 364/365 down to 343/365. Multiplied together they give a chance of about 0.4927 that everyone differs. The chance that two share a birthday is therefore 1 − 0.4927 = 0.5073, or 50.73%. That is why 23 is the smallest group above one half.
| People | Chance of a shared birthday |
|---|---|
| 10 | 11.69% |
| 20 | 41.14% |
| 23 | 50.73% |
| 30 | 70.63% |
Frequently asked questions
Why is the chance so high with only 23 people?
Because every pair of people is compared, not each person with one fixed birthday. Twenty-three people make 253 pairs, and each pair has about a 1 in 365 chance of matching, so the chances add up quickly.
What group size gives a 50% chance?
With 365 days the smallest group is 23 people, with a chance of about 50.73%. With 22 people the chance is about 47.57%, which is below one half.
Does a 366-day year change the answer?
Only a little. Counting 29 February as a birthday still gives 23 people for a 50% chance. The page offers the option because the usual model uses 365 days.
Are birthdays really spread evenly through the year?
No. Births vary by season and by day of the week, and an uneven spread makes a shared birthday a little more likely than the model says. The difference is small, so the group sizes above stay close to reality.
Why does a target of 100% give 366 people?
With 365 days, a 366th person must share a day with someone who came before, by the pigeonhole principle. So 366 is the smallest group certain to have a shared birthday.
Sources
Figures and text last checked on Tuesday, November 10, 2026.