52 people ≈ 97%
Double the “23 people” headline and you’re nearly guaranteed: 52 people is about a 97% chance of a shared birthday.
Quick math
Birthday Paradox Calculator — Probability of Shared Birthdays
Inputs
Tip: 23 people ≈ 50% chance with 365-day assumption.
This assumes birthdays are uniformly distributed across the chosen number of days.
Results
Set a group size and press Calculate.
Smallest Group for Common Thresholds
| Target Probability | Smallest Group Size |
|---|---|
| ≥ 50% | — |
| ≥ 75% | — |
| ≥ 90% | — |
| ≥ 99% | — |
Outputs are rounded for readability. Exact computations use high-precision log products to avoid underflow.
How the Birthday Paradox Calculator Works
The birthday paradox is a fun example of how probability can surprise us. This calculator estimates the chance that, in a group of people, at least two share the same birthday. Even though there are 365 possible days in a year, matches appear much sooner than most people expect. The tool makes the concept easy to explore without needing to do the math yourself.
At its core, the calculation asks: what is the probability that everyone in the group has a different birthday? If you know that value, you can subtract it from 1 to find the chance of at least one shared birthday. The model assumes each day of the year is equally likely, which keeps the math simple and is the standard approach for the classic paradox. You can also include leap day by switching to 366 days if you want to model February 29.
How to use it step by step
- Enter the group size (number of people).
- Choose whether to use a 365-day or 366-day year.
- Review the probability of at least one shared birthday.
- Check the threshold table to see how many people are needed for 50%, 75%, 90%, or 99% odds.
Real-world uses include teaching probability in classrooms, explaining why collisions happen in hashing, or understanding why duplicate events appear in large datasets. The same logic shows up in cybersecurity when discussing hash collisions and in analytics when checking for overlapping dates. It is also a great conversation starter for team events or workshops because the result feels counterintuitive.
For example, with just 23 people, the chance of a shared birthday is about 50%. By the time you reach 50 people, the probability is already above 97%. These numbers highlight how quickly combinations grow as group size increases. The calculator uses precise math internally so the results stay accurate even for large groups.
Birthday Paradox: FAQs
Why is it called a paradox?
Because the result feels surprising: our intuition often confuses the question “someone matches someone” with “someone matches me.” The calculator answers the former.
Does it assume birthdays are equally likely?
Yes. Real-world data varies by month and region, but the uniform assumption is standard for the paradox and keeps the math transparent.
What if the group size exceeds the number of days?
If n > D, then by the pigeonhole principle the probability is 100%.
Is my data private?
Yes. All calculations run locally in your browser; no data is uploaded or stored on a server.
5 Fun Facts about the Birthday Paradox
Months aren’t equal
September birthdays often dominate in real data, so real-world odds can be slightly higher than the uniform 365-day model suggests.
Real-world skew
Poker paradox cousin
The same math explains why pair collisions in hash functions happen sooner than intuition says—just like matching birthdays in a crowd.
Crypto parallel
Leap day barely moves it
Including 29 February nudges the 50% threshold from 23 to 24 people—a tiny shift despite the extra day.
Leap tweak
Three-way matches
The chance of at least three people sharing a birthday passes 50% around 88 people (365-day assumption).
Higher collisions