Birthday Problem Calculator
Compute the probability that at least two people in a group share a birthday.
About Birthday Problem Calculator
The birthday problem asks: in a group of n people, what is the probability that at least two share the same birthday? The result is surprisingly high — with just 23 people, the probability exceeds 50%. This counterintuitive result arises because we check all possible pairs, not just one pair.
Formula
P(match) = 1 − (365/365 × 364/365 × ... × (365−n+1)/365)
- n
- Number of people
Worked example
With the default values below, Birthday Problem Calculator returns P(at least one shared birthday) of 0.507297. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Number of Peopleaccepts 2–365
- 23
- Days in Yearaccepts 1–366
- 365
Results
- P(at least one shared birthday)
- 0.507297
- P(all unique birthdays)
- 0.492703
Step by step
- 1Values used
- 2Formula applied
- 3P(at least one shared birthday)= 0.507297
- 4P(all unique birthdays)= 0.492703
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Why is it called a paradox?
Because the result is counterintuitive: most people guess you'd need far more than 23 people for a 50% chance of a shared birthday. It's surprising because we're checking all pairs, and 23 people form 253 pairs.
How many people for 99% probability?
About 57 people are needed for a 99% probability of at least one shared birthday.