A gathering of simple wooden person figures beside an unmarked calendar. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Ask which birthday match counts
In a room of 23 people, the standard uniform 365-day model gives about a 50.7297% chance that at least two people share a birthday. It does not give a 50% chance that someone shares your birthday. Any-pair matching compares all 23 × 22 / 2 = 253 pairs. Matching a specific date asks a much narrower question.
Calculate the no-match event first
The first person can have any birthday. The second must avoid one occupied day, the third must avoid two, and so on. For 23 independent uniform birthdays, the no-collision probability is (365/365)(364/365)…(343/365). Subtracting that product from one gives the probability of at least one shared birthday. A direct sum of pair-match probabilities would count outcomes with multiple matching pairs more than once.
Compare a fixed-date event
If your birthday is fixed and 22 other independent people are sampled, the probability that at least one of them matches your date is 1−(364/365)²², approximately 5.86%. The exponent is 22 because your own birthday is not an additional random attempt to match itself. If 23 other people are sampled, use 23 instead. Keep the population description beside the formula.
Know the model before checking a classroom
The Birthday Paradox Calculator handles any-pair collisions and finds the minimum group size reaching a chosen target. It assumes equally likely buckets and independent draws; real birthdays have seasonal variation and leap-day complications. The result is not an exact forecast for a particular school or household. Read the identifier-space guide to apply the same occupancy calculation to uniformly generated codes. For the distinction between conditioning and changing the event, the Conditional Probability Calculator provides a separate joint-probability model.
Check the model behind your result
Use the Birthday Paradox Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26