How many people until two share a birthday? Astonishingly, just 23 for even odds. The classic result that breaks everyone's intuition.
Birthday ParadoxLive
smaller than you'd think
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How many people until two share a birthday? Intuition says hundreds, but it takes just 23 for better-than-even odds, and 70 makes it near-certain. The trick is that the number of pairs grows quadratically — 23 people form 253 pairs, each a chance to match. It is why hash collisions and cryptographic birthday attacks happen far sooner than expected.
Reading this result: Past the tipping point: 23 people form 253 pairs, and a match is now more likely than not (50.7%) — far fewer people than the 365 intuition expects.
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How it works
The surprise dissolves once you count pairs, not people: 23 people form 253 pairs, each a chance to match. The probability of at least one shared birthday climbs steeply, passing 50% at 23 and 99% by 70. The same quadratic effect drives hash collisions and the birthday attack in cryptography, where security scales with the square root of the key space.
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