Why 23 people is enough
The intuition that fails here is that people count people when the thing that matters is pairs. You are not asking “does anyone share my birthday” — that really does need about 253 people for even odds. You are asking whether any pair in the room matches, and 23 people contain 23 × 22 ÷ 2 = 253 pairs. The 253 turns up in both questions, which is the whole trick.
The calculation itself runs backwards, through the probability that everyone is different. The second person must avoid 1 date, the third must avoid 2, and so on, so the probability of no shared birthday is the product (1 − 1/365)(1 − 2/365)⋯(1 − (n−1)/365), and the answer is one minus that. Each factor is close to 1, but there are a lot of them, and they multiply.
Reading the working
The box above shows that product term by term, and also the standard approximation 1 − e−n(n−1)/2d, which comes from replacing each 1 − i/d by e−i/d. For 23 people in 365 days the approximation gives 0.5000 against an exact 0.5073 — close, and it gets closer as the space grows, which is why nobody bothers with the exact product for hash sizes.
The same formula is a security calculation
Set the space to 2¹²⁸ and the calculator answers a different question: how many random UUIDs can be generated before two of them collide? Because the number of people needed scales as √d, a 128-bit space gives collision resistance of only about 2⁶⁴ — which is why a hash function advertised as “128-bit” is described as offering 64 bits of collision resistance, and why birthday attacks halve the effective strength of every hash. The rule of thumb for the 50% point is n ≈ 1.1774·√d, and the calculator uses that as the starting estimate before refining it.
Where the exact product stops working
For a large space the exact product has too many terms to multiply one at a time, and each factor is so close to 1 that ordinary floating point loses the difference entirely. This page computes the product in log space with log1p, which keeps the precision for small terms, and falls back to the asymptotic form when the number of people passes a few million. The working box always says which of the two produced the number on display.
Three people sharing is a different problem
There is no neat product for “at least three people share a date”, because the events overlap in ways that do not factor. The known answer for 365 days is about 88 people for even odds — roughly four times the pair case, not one and a half times. Move the slider and the page switches to simulation, which is honest about being an estimate rather than pretending to a precision it does not have.