How many people need to be in a room before two probably share a birthday?
Ignore leap years and assume birthdays are spread evenly across the year. Most people guess you'd need close to 183 people in a room before it's likely two share a birthday. How many people does it actually take before the odds tip past 50/50?
Reveal the answer
Just 23. The trick is that you're comparing every pair of people, not just one person against everyone else, and the number of pairs grows much faster than the number of people: 23 people already give you 253 chances for a match. By 70 people, the odds of a shared birthday exceed 99.9%.
— classical probability theory, Birthday problem — a standard result in combinatorics