Birthday Problem Calculator (Probability of a Match)
What is the probability that at least two people in a group of n share the same birthday?
Result
P(shared birthday)
50.7297%
P(all different)
49.2703%
50% crossover at
23 people
Group size needed for ≥50% probability
Possible pairs
C(23,2) = 253
Concept
The birthday problem is a classic probability paradox — most people guess far too high before seeing the answer. In a group of just 23 people, the probability of at least two sharing a birthday exceeds 50%.
The complement trick: it's much easier to calculate the probability that everyone has a different birthday and subtract from 1. Each new person must avoid all birthdays already taken.
Assumptions: 365-day year, uniform birthday distribution. Real birthdays are not uniformly distributed — certain months have higher birth rates — so the real crossover is actually lower than 23.
Formula
Variables
n- Number of people in the group.
365- Days in a year — can be changed for non-standard calendars.
P(no match)- Probability that all n people have distinct birthdays.
P(at least one match)- Probability that at least one pair shares a birthday = 1 − P(no match).