Birthday paradox
Drag the room size and watch the probability climb. The curve crosses 50% earlier than almost anyone guesses, and tops 99% before you fill a single bus.
P(at least one shared birthday)
50.73%
With 23 people in the room. Expected matching pairs: 0.69.
You only need 23 people for a 50% chance, and 57 for 99%.
Same formula at common headcounts. The 50% crossover sits between 22 and 23.
| People (N) | P(shared) | Expected pairs |
|---|---|---|
| 5 | 2.71% | 0.03 |
| 10 | 11.69% | 0.12 |
| 15 | 25.29% | 0.29 |
| 20 | 41.14% | 0.52 |
| 23 | 50.73% | 0.69 |
| 30 | 70.63% | 1.19 |
| 40 | 89.12% | 2.14 |
| 50 | 97.04% | 3.36 |
| 70 | >99.9% | 6.62 |
| 100 | >99.9% | 13.56 |
The birthday paradox is not a paradox in the logical sense. It is counterintuitive because we instinctively count people rather than pairs. With 23 people you do not have 23 chances at a match. You have 253. That is C(23, 2) = 253 distinct pairs, and each pair has a 1 in 365 chance of sharing a date. The probability that none of those 253 pairs match drops below half almost exactly at N = 23.
The clean way to compute it is to ask the opposite question: what is the chance that everyone has a different birthday? The first person picks any day (365/365). The second avoids it (364/365). The third avoids two days (363/365). Multiply the fractions together for N people and subtract from 1. The product shrinks fast because each extra person multiplies in a smaller fraction. By N = 50 the chance of a match is already 97%, and by N = 70 it is over 99.9%.
You can sanity check this against the real world. An NFL roster holds 53 active players, so any given team has roughly a 98% chance that two players share a birthday. NBA teams carry 15 players and land around 25%. A typical classroom of 30 students sits near 70%. Once you have a feel for the pair count C(n,2) growing as n squared, the surprise disappears. The math is honest about how much faster pairs grow than people.
These reference values come from the same model the slider uses: 365 equally likely birthdays, probability computed from the chance that every person misses every earlier person's date. The expected pairs column is C(n, 2) divided by 365, which explains the shape of the curve before you ever drag the slider.
| People in the room | P(shared birthday) | Expected matching pairs |
|---|---|---|
| 10 | 11.7% | 0.12 |
| 15 | 25.3% | 0.29 |
| 23 | 50.7% | 0.69 |
| 30 | 70.6% | 1.19 |
| 40 | 89.1% | 2.14 |
| 50 | 97.0% | 3.36 |
| 70 | 99.9% | 6.62 |
Three simplifications sit behind the numbers. First, every date is treated as equally likely, while real birth registries show seasonal peaks; clustering only pushes match probability up, never down. Second, February 29 is ignored, which nudges probabilities up by a sliver because there are 365 buckets instead of 366. Third, birthdays are assumed independent, so a pair of twins in the room breaks the model outright: they are a guaranteed match that the formula never priced in. For quiz-night purposes none of these gaps matter. The crossover stays at 23 whether you run the clean model or feed in messy registry data.
Any system that assigns random values from a fixed pool runs into this curve. Collisions become likely once the number of items approaches the square root of the pool size, which is why 23 people, close to the square root of 365 scaled by a small constant, is enough for birthdays. Cryptographers call the adversarial version a birthday attack: finding two inputs that hash to the same output takes far fewer attempts than finding an input matching one specific output. The same reasoning tells you how many records a database can hold before two random session tokens collide, and why doubling a token's length squares the safety margin rather than merely doubling it.
No. It uses the textbook model of 365 equally likely birthdays. Leap-day births are rare enough that including them moves the headline numbers by only a fraction of a percentage point.
That is a different problem from the paradox. Matching one fixed date follows 1 - (364/365)^(n-1), which does not pass 50% until about 253 other people are in the room, roughly ten times the 23 needed for any two people to match.
23 is the smallest room size where the probability crosses one half. The chance of a shared birthday at 23 people is about 50.7%, while at 22 people it is still just under 50%.
Each extra person multiplies the no-match chance by a fraction close to 1, so the remaining gap shrinks slowly. In a 365-day model a match is only guaranteed outright once 366 people are present, by the pigeonhole principle.
Not perfectly. Birth records cluster in some months, and any clustering makes collisions slightly more likely than the uniform model predicts. The 365-day assumption is therefore a mild underestimate, so the surprising result holds in real rooms too.