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Four discs in a row with six arcs hanging below them Four filled discs stand in a row along the top. Below them six curved lines hang downwards, joining every possible pair of discs: three shallow arcs between neighbours, two deeper ones skipping a disc, and one deep arc spanning the whole row. Four things make six pairs.

Count the Pairs, Not the People

About 11 minutes

Twenty-three children in a room. There are 365 days in a year. What are the chances that two of them have the same birthday?

Most people say something small — one in ten, maybe less. Twenty-three against 365 does not look like much of a contest.

The real answer is 50.7%. Slightly better than a coin flip. In a class of thirty it climbs to 70.6%, and by fifty it is 97%.

Nothing here is a trick. The number is genuinely that big, and the reason is that almost everybody counts the wrong things.

The question you answered was not the question asked

When you pictured twenty-three children and 365 days, you almost certainly pictured yourself standing among them, looking for a match.

That is a different question, and its answer really is small. You have 22 other children to check against, so there are 22 chances, and the chance that at least one of them matches you is about 5.9%. One in seventeen. Your instinct was right about that.

But nobody asked about you. The question is whether any two of them match — and that includes every pair that does not contain you at all.

How many pairs is that

Line the children up and add them one at a time, counting the new pairs each arrival makes.

Six children in a row and the fifteen pairs between them Six numbered discs stand in a row. Below them, fifteen curved lines drawn in a second colour join every possible pair of discs. Arcs between neighbouring discs are shallow and arcs spanning the whole row are deep. Each new child added to the row makes one more arc than the child before did: the second child makes one, the third makes two, and so on up to the sixth, who makes five. Six children make fifteen pairs. 123456 discs are children arcs are pairs — six children already make fifteen of them

The discs are children. The arcs are pairs. Choose an arrival to see how many new pairs walking in creates.

  • The 2nd child — makes 1 new pair, with the child already there. Total pairs: 1.
  • The 3rd child — pairs with both of the others, so 2 new arcs. Total: 3.
  • The 4th child — pairs with all three, so 3 new arcs. Total: 6.
  • The 5th child — pairs with all four, so 4 new arcs. Total: 10.
  • The 6th child — pairs with all five, so 5 new arcs. Total: 15. Six children already have more pairs between them than there are children in the room.

Each new arrival shakes hands with everyone already there, so the twenty-third child adds 22 pairs. Add them all up:

1 + 2 + 3 + … + 22 = 253

Twenty-three children. Two hundred and fifty-three pairs. That is the number that should have been compared with 365, and suddenly the answer being about half stops feeling strange at all.

This is the same move as part 1. A total of seven is not one thing, it is six rolls wearing one label. A room of twenty-three is not twenty-three chances, it is two hundred and fifty-three.

Working it out properly

There is a rough version and an exact version, and the rough one is worth doing first because it explains the size.

Rough: 253 pairs, each with about a 1 in 365 chance of matching. 253 ÷ 365 is about 0.69. So something like 69%.

That is too big, and it is too big for an honest reason: the pairs overlap. If Amy matches Ben and Ben matches Chris, then Amy matches Chris automatically — that third pair was not a fresh chance. Adding up overlapping chances always overshoots.

Exact: turn the question round and work out the chance that everybody is different, which has no overlapping to worry about.

Multiply all twenty-three fractions together and you get 0.493. So the chance that everybody is different is 49.3%, and the chance that somebody is not is 100 − 49.3 = 50.7%.

Each fraction on its own is very close to 1. Twenty-three of them multiplied together is not.

Children Pairs Chance two share a birthday
5 10 2.7%
10 45 11.7%
23 253 50.7%
30 435 70.6%
41 820 90.3%
50 1,225 97.0%

Twenty-three children are in a room and you are one of them. Which is more likely?

The same mistake, three times now

Look back at what has actually gone wrong in this book, because it has been one mistake each time wearing a different coat.

Seven beat twelve because a total is not a roll — six rolls hide behind one label. Five heads in a row felt rarer than HTHT because a description is not a result — one description can cover six endings or one. And a room of twenty-three has 253 pairs because a person is not a chance.

Every time, the fix was the same: stop counting the things you can see, and count the things that can actually happen.

Which leaves one last place to point this. Every chapter so far has been about being fooled by dice, coins and calendars — objects, out there, that nobody claims are clever.

The thing doing the fooling has been inside your head the entire time. Can you catch it?