About the Birthday Paradox Calculator
The birthday paradox says that in a group of just 23 people there is a better than even chance that two of them share a birthday. It feels wrong, which is why it is called a paradox, but the maths is simple: a group of 23 contains 253 pairs of people, and each pair has a small chance of matching. This calculator gives the chance for any group size, or turns the question around and tells you how many people you need for a chosen probability. You can change the number of possible days, which is useful for related puzzles such as hash collisions.
Updated October 2026
How to use the Birthday Paradox Calculator
- 1Choose to find the chance for a group size or the group size for a chance.
- 2Enter the number of people, or the chance you want as a percent.
- 3Leave the possible birthdays at 365, or set another number.
- 4Read the probability or the group size in the result box.
- 5Compare it with the chance that everyone's birthday is different.
Numbers to remember
- 10 people: about 11.7 percent.
- 23 people: about 50.7 percent.
- 30 people: about 70.6 percent.
- 50 people: about 97 percent.
- 70 people: about 99.9 percent.
Using it for class or party demos
A classroom of 30 is the classic demonstration. Ask students to write down their birthdays, then check for matches; roughly seven times out of ten there will be at least one. Football squads, with about 23 players on the pitch plus the referee, are another favourite example.
The reason the answer feels so surprising is that people picture themselves being matched, rather than every pair in the room. In a group of 70 there are 2,415 pairs, and each of them is a fresh chance for a match. The count of pairs grows with the square of the group size, while the number of possible days stays fixed, so collisions arrive much sooner than intuition expects.
Changing the number of days
Replace 365 with any other number to model similar problems: 52 for shared birth weeks, 12 for birth months, or 1,000,000 for random ID collisions. For other chance problems try the Probability Calculator and the Binomial Distribution Calculator, and for counting orders see the Permutation and Combination Calculator.
Features
- Chance for any group size
- Group size for any target chance
- Adjustable number of possible days
- Shows the number of pairs compared
- Chance that all birthdays differ
- Exact calculation, not an approximation
Frequently asked questions
Why are 23 people enough for a 50 percent chance?
Because you are not comparing everyone with you, but every pair. Twenty-three people form 253 pairs, and the chance that nobody matches works out to about 49.3 percent, which leaves a 50.7 percent chance of at least one match.
How is the probability calculated?
Find the chance that all birthdays are different by multiplying 365/365 × 364/365 × 363/365 and so on for each person, then subtract from one. The calculator does this exactly with no shortcut approximation.
How many people guarantee a shared birthday?
With 366 people there must be a match, even counting 29 February, which is the pigeonhole principle. A 99.9 percent chance needs only 70 people.
Does the paradox apply to computer security?
Yes. The same maths explains why hash functions with n possible values start to produce collisions after roughly the square root of n inputs. That is the idea behind the birthday attack on cryptographic hashes.
Are birthdays really equally likely?
Not quite. Some times of year have more births, which makes a shared birthday slightly more likely than the calculator says. Leap days and twins also have small effects.
What about the chance of matching my own birthday?
That is a different question. For a specific date you need about 253 people for a 50 percent chance, because only comparisons with you count. The paradox is about any two people matching.