THE PARADOX
Any two, not you
With 23 people there are 253 possible pairs. Each pair is unlikely to match, but there are a lot of them.
See the sumsTHE BIRTHDAY PARADOX
23 people.
Even odds.
Two people in a room of 23 probably share a birthday.
Check the odds for your room
253pairs in 23
- Count pairs, not people
- 23 for even odds
- Late September is busiest
- Christmas is quietest
It sounds wrong, because 23 is a long way short of 365. The trick is that every person can match every other person, not just you. Here is why the numbers work, plus what the national records say about the busiest and quietest birthdays of the year.
Plain English, with the sources linked.
Updated
THE SHORT ANSWER
Count the pairs.
The surprise is not the birthdays.
It is how fast the pairs add up.
THE BUSIEST DAYS
Late September
In England and Wales, 26 September was the most common birthday over two decades of records.
Why September?
TRY THE NUMBERS
How big is your room?
Pick a group size and see the exact chance of a shared birthday, from a dinner party to a village hall.
Check the oddsA QUICK ROOM CHECK
How many people
are in the room?
Pick a group size. The figure is the chance that at least two people in it share a birthday, worked out for 365 equally likely days and ignoring 29 February.
Real birthdays are not spread perfectly evenly, and an uneven spread makes a shared birthday slightly more likely, not less.
TWO NUMBERS WORTH KNOWING
Small groups,
busy calendars.
One is pure arithmetic.
The other comes from 20 years of birth records.
253 pairs
Possible pairs among 23 people
Any one pair shares a birthday only 1 time in 365. But 23 people can be paired 253 ways, and across all of them the chance of no match at all drops to just under half.
THE PARADOX · ARITHMETIC
2,000 vs 1,800
Births on an average 26 September
If births were spread evenly, England and Wales would expect about 1,800 a day. On 26 September the average was around 2,000, and 8 of the 10 most popular birthdays fell in late September.
THE CALENDAR · ONS, 1995 TO 2014
The two ideas are linked. Because some dates are busier than others, a room of real people is a little more likely to contain a shared birthday than the even-calendar maths suggests.
TRY IT AT A PARTY
Four ways to
test the paradox.
Count the room first
With 23 or more people, it is a fair bet that two share a birthday. With 40 or more, it is a strong one.
Go round the calendar
Call out the months one by one and ask people to shout when they hear theirs. Then narrow it down by date.
Do not bet on your own date
The odds that someone shares your birthday are far lower: about 6% among 22 other people. You would need 253 others for even odds.
Try the near-miss version
In a smaller group, count birthdays on the same day or one day apart. With 14 people, that is already close to even odds.
BIRTHDAY FACTS
Five things worth
celebrating.
Forty people, nine in ten
Australia's Chief Scientist notes that with 23 people the chance of a shared birthday is just over 50 per cent, and with 40 people it rises to around 90 per cent.
Near misses come even sooner
With only 14 people there is about an even chance that two have birthdays on the same day or one day apart. It takes 88 people for an even chance that three share one birthday.
September babies, Christmas conceptions
The ONS found 26 September was the most common birthday in England and Wales from 1995 to 2014. It links the late-September peak to more babies being conceived around Christmas.
The quietest days are bank holidays
6 of the 10 least common birthdays fall over Christmas and New Year, when fewer births are planned. A study of 29 million births found the quietest single day was Boxing Day 1979, with 1,123.
The song started as a morning greeting
The tune of Happy Birthday to You began as Good Morning to All, published in 1893 by the sisters Mildred and Patty Hill. A 2016 court settlement put the birthday song in the public domain.
QUICK QUESTIONS
Birthdays, answered.
Short answers to the questions people ask most.
How many people do you need for two to share a birthday?
Just 23. With 23 people there are 253 possible pairs, and the chance that at least one pair shares a birthday is about 50.7%, assuming every day is equally likely and ignoring 29 February. With 40 people it is about 89%.
Why does the birthday paradox feel wrong?
Most people picture someone matching their own birthday. The paradox is about any two people matching. The number of possible pairs grows much faster than the number of people: 23 people make 253 pairs.
What are the odds someone shares my birthday?
Much lower. In a group of 23, the chance that one of the other 22 shares your particular birthday is only about 6%. You would need 253 other people for that to reach an even chance.
What is the most common birthday in England and Wales?
The Office for National Statistics found that 26 September was the most popular birthday from 1995 to 2014, with about 2,000 births on an average 26 September against about 1,800 on an average day.
What is the least common birthday?
29 February has the lowest total because it only comes round once every four years. Among ordinary dates, the quietest days cluster around Christmas and New Year: 6 of the 10 least popular birthdays fell in that period.
Where the facts come from
- The birthday problem (2011)Australia's Chief Scientist
- Using probability to understand the birthday paradoxScientific American
- How popular is your birthday? (1995 to 2014)Office for National Statistics
- The most popular birthday in England and Wales (2017)Significance, Royal Statistical Society
- Copyright Lore: Happy Birthday to You (2010)US Copyright Office
- The forgotten sisters behind Happy Birthday to YouSmithsonian Magazine