The Birthday Problem
Folkloric; Richard von Mises gave the first formal treatment in 1939 · early 20th century
The puzzle
How many people in a room before the probability that two share a birthday exceeds 50%?
Note
Twenty-three. With seventy people, it’s 99.9%. The result feels wrong because most people quietly substitute the harder question — what is the probability someone matches me — which has answer ~253 for 50%. Your birthday isn’t fixed; we’re counting any pair, and the number of pairs in a group of 23 is 253. Not a paradox in the strict logical sense — nothing contradicts — but it stress-tests the same human-reasoning gap that Monty Hall does: we under-count the structure of any claim relative to this specific claim. Cryptographers exploit it routinely under the name “birthday attack”: a hash function with N possible outputs can be expected to collide after roughly √N inputs, not N.