Birthday Paradox Probability Calculator
The famous birthday paradox shows that duplicate birthdays appear much sooner than intuition expects. Enter the number of people in a group to calculate the probability that at least two share the same birthday under idealised assumptions.
A theoretical calculation that ignores cultural naming conventions, hospital birth patterns, or leap-day birthdays.
Examples
- 23 people ⇒ 50.73% chance of a shared birthday
- 40 people ⇒ 89.12% chance of a shared birthday
- 60 people ⇒ 99.41% chance of a shared birthday
FAQ
Why does the probability rise so quickly?
Every new person can match with all earlier birthdays, so the number of possible pairs grows rapidly even in small groups.
What formula is being used?
The tool computes 1 − Π_{k=0}^{n−1}((365 − k)/365), where n is the number of people in the room.
Does the calculator consider leap years or seasonal birth patterns?
No, it assumes a perfectly uniform distribution. Real-world data with seasonal birth peaks make matches slightly more common.
Can I adapt this to other matching problems?
Yes—any situation with equally likely outcomes, such as hash collisions or randomly assigned IDs, can reuse the same logic.
What group size guarantees a match?
With 366 people the probability is 100% by the pigeonhole principle, but the odds already exceed 99% around 57 people.
Additional Information
- The calculation multiplies the probability that every birthday is unique and subtracts the result from 1.
- Assumes 365 equally likely birthdays and independent dates—real-world clustering around certain months can increase the odds further.
- February 29 is excluded; include a 366th day if you want to account for leap-year birthdays.
- Use the result to plan icebreakers, marketing giveaways, or demonstrations about probability and data collisions.