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ELI5 Yahaaa

A 50.7 per cent chance

In a Room of Twenty-Three, Two People Share a Birthday

It feels far too few. The mistake is counting people when you should be counting pairs.

What you picture 23 people against 365days — surely notWhat actually matters 253 pairs of people,each with a chance
Both are true descriptions of the same room. Only one predicts the answer.

  1. Step 1 of 5

    The instinct is to compare 23 with 365, and that is the wrong comparison.

    Which question are youactually asking?Someone matches me: 22 chancesAny two of us match: 253 chances
    Same room, same 23 people. The second question has eleven times the chances.

    Twenty-three people, three hundred and sixty-five days: it feels hopeless. But nobody asked whether someone shares a birthday with you. The question is whether any two people in the room match, and that is a different question with a much bigger answer.

    For someone to match your birthday specifically, you need about 253 people in the room before it becomes more likely than not. That is the number people's instincts are actually reaching for.

  2. Step 2 of 5

    Count the pairs instead.

    Pairs made by thefirst six peoplealoneAnd the countkeeps climbingwith each arrivalTwenty-threepeople: 253 pairsin total

    Person one can pair with 22 others. Person two has 21 new partners left, person three has 20, and so on. Add them all up and 23 people make 253 different pairs — each one a separate opportunity for a match.

  3. Step 3 of 5

    The number of pairs grows much faster than the number of people.

    10 people45 pairs20 people190 pairs23 people253 pairs — thetipping point30 people435 pairs

    Double the people and you roughly quadruple the pairs, because every new arrival pairs with everyone already there. Ten people make 45 pairs; twenty make 190; thirty make 435. This is why the answer arrives sooner than instinct expects.

  4. Step 4 of 5

    The neat way to calculate it is backwards.

    1 Person 2must miss 1date364/3652 Person 3must miss 2dates363/3653×Multiplyevery one ofthemEach factoris slightlyunder 14 By person 23the productis 0.493So a matchhas 50.7%probability

    Work out the chance that everybody is different, then subtract. The second person must dodge one date, so 364 chances in 365. The third must dodge two, so 363 in 365. Multiply all those together and by the 23rd person the product has fallen below one half.

  5. Step 5 of 5

    It stops being surprising once you have seen it.

    23 people50.7% — thefamous number30 people70.6%40 people89.1%50 people97.0%70 people99.9%
    It climbs steeply, because the pairs do.

    By 50 people the odds are about 97 per cent. By 70 it is 99.9. And it is not a trick of uneven real-world birthdays — real birth dates cluster slightly, which only makes matches more likely, not less.

The short version

Twenty-three people make 253 pairs, and it is the pairs that get the chances — which is why the odds of a shared birthday pass one half far sooner than instinct suggests.

Try it yourself

Try it on any group of about 25 — a class, a team, a bus queue. It works more often than not, and the surprise on people's faces is most of the fun.

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