Birthday Paradox Calculator — Group Size for a Shared Birthday

Formula

How this is calculated

The birthday paradox asks: in a room of how many people is there at least a 50% chance that two share a birthday? The answer — just 23 — surprises almost everyone because people instinctively compare themselves to one specific other person, rather than counting all pairs. With 23 people there are 253 unique pairs (23 × 22 ÷ 2), and the probability that none of those pairs match is less than one half.

This calculator solves the inverse problem: given a desired probability level and the number of possible days, it finds the smallest group size n for which P(shared) ≥ target. Internally it iterates n from 2 upward, computing log P(all distinct) = Σᵢ₌₀ⁿ⁻¹ ln((d−i)/d) and setting P(shared) = 1 − exp(log P(all distinct)). Using logarithms avoids floating-point underflow that would occur when multiplying many small fractions directly. The loop stops at the first n that meets the target.

Assumptions: birthdays are independent and uniformly distributed over d days. In reality, birth rates peak slightly in summer months, making coincidences marginally more common than this model predicts. Both n and d are positive integers; if n exceeds d, the pigeonhole principle guarantees a collision (P = 1). The probability curve in the output shows how steeply P rises as the group grows — it crosses 99% at just 57 people and 99.9% at 70 people for d = 365.