The birthday paradox: why 23 people is enough

Put 23 people in a room and it is more likely than not that two of them share a birthday: 50.7%. With 30 it is 70.6%, and with 70 it is 99.91%. It is easy to confuse with a different question — whether someone shares your birthday — which has a much smaller answer.

How it works, one person at a time

It is easier to work out the chance that nobody shares a birthday, then subtract that from one. Let people walk in one at a time. The first can have any birthday. The second must avoid the first’s: 364 of 365 days will do. The third must avoid two taken days, so 363 of 365, and so on. Every one of them has to dodge, so the chances multiply.

So the chance nobody matches is 29.4%, and the chance someone does is one minus that: 70.6%. To check it on any calculator, type:

1 − (365 × 364 × … × 336) ÷ 365^30

This assumes 365 equally likely birthdays and ignores 29 February. Both assumptions are tested against real births further down.

Your birthday versus anyone’s

The chance that one of the other 29 people in a room of 30 shares your birthday is only 7.6%. You are in 29 of the 435 pairs; the other 406 don’t involve you at all, and any of them can produce the match. Each of the others misses your birthday with probability 364/365, so for an even chance of a match with you, you need 253 other people.

Three people on one day

In a group of 30 the chance that three people share a birthday is 2.9%, about 1 in 35. An even chance of three takes 88 people, four takes 187 and five takes 313.

Real birthdays are not spread evenly

The textbook answer assumes every day is equally likely. It is not. These are the 62,187,024 births in the United States from 2000–2014, as counted by the Social Security Administration, by date of the year.

Shaded by rank: the fainter the square, the fewer the births. Point at or tap a day to read it.

Each figure is that date’s births as a share of an average day’s.

Christmas Day is the rarest birthday in the calendar, with 57% of an average day’s births. The rest of the five rarest sit on or beside Christmas, New Year and the Fourth of July, and four of the ten rarest fall between 22 and 28 November, the week Thanksgiving always lands in. At the other end, seven of the ten most common birthdays are in September. The data shows when births happen, not why, so this page does not guess. 29 February, judged on the years it exists, sees 92% of an average day’s births.

Does it change the answer? Barely, and only upwards

With real birthdays, the chance of a match in a group of 30 is 70.7% against 70.6% on paper — 0.06 percentage points higher. A calendar this lumpy moves the answer this little, and 23 people is still where it passes one half.

It can only move upwards, and the reason fits in a paragraph. Pick any two days, one more popular than the other. Leave every other day alone and the chance that nobody matches depends on those two days through just two quantities: their combined share, which is fixed, and their shares multiplied together. The bigger that product, the likelier it is that nobody matches — and for a fixed total the product is biggest when the two shares are equal. So evening out any two days makes a match less likely. Keep doing it and you arrive at a perfectly even calendar, where a match is as unlikely as it can be. Any real calendar must beat it.

Strictly, that compares like with like: the real 366 dates, leap day included, against 366 equally likely ones, which give 70.5% for 30 people. The real figure is above it at every group size from 2 to 100, as it has to be. And a separate count by the CDC National Center for Health Statistics, covering 1994–2003, agrees with it to within 0.003 percentage points at every one of those group sizes.

Questions people ask

How many people do you need for two to share a birthday?

23. With 23 people the chance that at least two share a birthday is 50.7%, just over half; with 22 it is 47.6%.

What are the odds two people in a room of 30 share a birthday?

70.6%, assuming 365 equally likely birthdays. Using real US birth data instead it is 70.7%.

How many people do you need for someone to share your birthday?

253 other people for a better-than-even chance. Matching one particular date is a much harder target than matching any two people, which is why the two answers are so far apart.

Does the birthday paradox work with real birthdays?

Yes, slightly better. Real birthdays are not spread evenly, and an uneven spread can only raise the chance of a match. With 23 people it is 50.8% on real US birth data against 50.7% on paper.

How the site works out its numbers is on the methodology page.