Birthday Paradox Calculator
Use this tool to calculate how likely it is that at least two people share the same birthday in a group.
Find group size for a target probability
Tip: With 23 people and 365 days, the probability is about 50.73%.
What is the birthday paradox?
The birthday paradox is a famous probability result: in a surprisingly small group of people, the chance that two people share a birthday gets large very quickly. Most people guess you need over 100 people to get a 50% chance. In reality, you only need 23 people (assuming 365 equally likely birthdays).
How to use this calculator
1) Calculate probability from group size
- Enter the number of people in the group.
- Set the number of possible birthdays (default is 365).
- Click Calculate Probability.
2) Calculate required group size from target probability
- Enter a target probability, such as 90% or 99%.
- Click Find Minimum Group Size.
- The tool returns the smallest group size that meets or exceeds your target.
The formula behind the result
It is easier to compute the opposite event first: the probability that no one shares a birthday.
For a group of n people and d equally likely days:
P(no match) = (d/d) × ((d-1)/d) × ((d-2)/d) × ... × ((d-n+1)/d)
Then:
P(at least one match) = 1 - P(no match)
This calculator uses that complement approach, which is numerically stable and accurate for practical inputs.
Common benchmarks (365-day year)
- 10 people: ~11.69%
- 20 people: ~41.14%
- 23 people: ~50.73%
- 30 people: ~70.63%
- 40 people: ~89.12%
- 50 people: ~97.04%
- 57 people: ~99.01%
Assumptions and limitations
This model is intentionally simple. It assumes:
- All birthdays are equally likely.
- Birthdays are independent from person to person.
- No distinction for leap day unless you change days to 366.
Real birth data is not perfectly uniform, but these assumptions are usually good enough for learning and estimation.
Why this matters beyond birthdays
The same math appears in computing and security:
- Hash collisions: how likely two inputs produce the same hash value.
- Random ID systems: collision risk when generating many identifiers.
- Cryptography: birthday attacks against functions with limited output size.
So while this starts as a fun puzzle, it also gives intuition for collision risk in real-world systems.
Quick FAQ
Why is it called a paradox?
It is not a logical contradiction. It is “paradoxical” because the true probability feels much higher than intuition expects.
What if I only care whether someone matches my birthday?
That is a different probability and much smaller. For n people total (including you), it is
1 - (1 - 1/d)^(n-1).
What if the group size is larger than the number of possible days?
Then a shared birthday is guaranteed by the pigeonhole principle, so the probability is 100%.
Bottom line
The birthday paradox is a great reminder that human intuition often struggles with probability. Use the calculator above to explore different group sizes and see how quickly collision probability rises.