Why is the answer so much higher than most people guess?+ā
Because the question is about any pair sharing a birthday, not about someone sharing your specific birthday. With n people there are n(nā1)/2 pairs to check ā 23 people make 253 pairs, and each pair has its own small chance of a match. Those 253 chances add up fast, which is why the probability crosses 50% at just 23 people, not the 183 (half of 365) that intuition suggests.
How is the exact probability calculated?+ā
It's easier to compute the complement ā the probability that nobody shares a birthday ā and subtract from 1. The first person can have any birthday (365/365 chance of 'no conflict so far'), the second must avoid the first's (364/365), the third must avoid both (363/365), and so on. Multiply all those fractions together for the probability of no shared birthday, then subtract from 1.
Does this assume birthdays are evenly spread across the year?+ā
Yes, and it ignores leap years (treating the year as 365 days) and real-world effects like seasonal birth clustering. The real-world probability is actually very slightly higher than this simplified model predicts, since uneven birthday distribution increases collision chances rather than decreasing them.
At what room size does it become a near-certainty?+ā
By 70 people, the probability is above 99.9%, and by 100 people it's effectively certain ā the curve on this page flattens out near 100% well before you reach a full classroom.