The birthday paradox: why 23 people are enough to share a birthday
The birthday paradox explained: count the pairs, not the people
In a group of 23 people, the chance that someone shares your birthday is small. The chance that some two people share a birthday is better than even. The whole difference is in what you count.
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1
You look through the 22 other passports. Nobody was born on 2 October. That is normal: each person has 1 chance in 365 to share your birthday. With 22 people, that gives you only about 6 chances in 100.
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2
Rono smiles. His bet never needed you. Any two travellers would do: Miva and the man from the back row, or two people you never talked to.
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3
So count the pairs instead. Take four travellers. Each of the four can match the three others. That makes 12, but it counts every pair twice: Miva with you is the same pair as you with Miva. So there are 6 pairs. Every pair is one more chance for a match.
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4
When a new traveller joins, they make a pair with everyone already there. The 6th person adds 5 new pairs. The 23rd person adds 22.
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5
Add them all up, and 23 people make 253 pairs. Your own birthday had 22 chances to match. The whole group had 253.
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6
Each pair matches 1 time in 365. Add the chances of all 253 pairs, and you get 69 in 100. Yet the real chance is 51 in 100, and the count behind it also lets forgers fake signatures. Where did the rest of the chances go?
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Quick check
Six friends sit down for dinner. How many pairs at the table could share a birthday?
Show the answer
15 pairs. Each friend who sits down makes a pair with everyone already seated: 1 + 2 + 3 + 4 + 5 = 15.