Advanced options
The famous result: just 23 people gives a better-than-even chance of a shared birthday, because what's being counted is PAIRS of people (253 of them at 23), not comparisons against your own birthday. Assumes 365 equally likely birthdays: real birth-date clustering (hello, late September) pushes the odds slightly higher still.
What The Birthday Paradox Calculator Works Out
- Chance two share a birthday: computed on the complement, multiplying the shrinking probabilities that each new arrival misses everyone before them (365/365 x 364/365 x 363/365 and so on) and subtracting from one. A mere 23 people gives 50.7%, 30 people passes 70%, 41 passes 90% and by 70 a match is a 99.9% certainty.
- Why it feels wrong: the question counts pairs, not people. 23 people contain 253 pairs, each an independent-feeling roll at a 1-in-365 match. Intuition fails by answering a different question: "does anyone share MY birthday" genuinely does need 253 people for even odds.
- Landmarks: the probabilities at the classic group sizes, so you can settle the bet before the register is called. Real birthdays are not uniform (September's bulge and the scheduled-caesarean weekday effect), so actual match rates run slightly above the textbook figure.
- What has to match (advanced): swap birthdays for birth months (12), birth weeks (52), the weekday of birth (7) or any custom number of categories, such as the 1,000 possible first-three-digits of a PIN or the 366 days including 29 February. The same arithmetic prices DNA-database coincidences, duplicate identities in a million records and any "what a coincidence!" story: coincidences are cheap at scale.
- Chance someone shares YOUR birthday (advanced): the question people think they are asking: 1 - (364/365) to the power of the other people in the room. At 23 people it is only 5.9%, and the sub-line gives the room size needed for your target.
- People needed (advanced): the smallest group that reaches your chosen probability of some shared birthday. 23 for 50%, 32 for 75%, 41 for 90%, 57 for 99%.
- Expected matching pairs (advanced): pairs divided by categories, the average number of coincidences you would find. Once this passes about 0.7 a match is more likely than not.
- Square-root shortcut (advanced): the approximation 1 - e^(-pairs/categories) used in cryptography, where the "birthday attack" finds hash collisions in roughly the square root of the brute-force effort, so a 128-bit hash offers only 64 bits of collision resistance; that arithmetic retired MD5 and SHA-1 from signatures. Matches become likely once the group approaches 1.18 times the square root of the categories, which is where 23 comes from.
- Probability table (advanced): group sizes against the chance of any shared category and the chance one matches yours, so the two questions sit side by side.