Birthday paradox calculator

Enter the number of people. The chance is computed exactly with the product formula, not an approximation, and works up to the point where a match is certain.

Number of peopleA whole number from 1 up.

Days in the year365 by default. Use another number for a different calendar, or a die with that many sides.

Chance of at least one shared birthday50.7297%

Chance that all birthdays differ: 49.2703%
As odds: 1 in 1.971
People needed for 50%: 23
People needed for 99%: 57
5 people: 2.71356%
10 people: 11.6948%
15 people: 25.2901%
20 people: 41.1438%
23 people: 50.7297%
30 people: 70.6316%
40 people: 89.1232%
50 people: 97.0374%
60 people: 99.4123%
70 people: 99.916%
100 people: 100%

Runs in your browser: nothing you enter is sent anywhere.

How to use it

Type the number of people in the group.
Leave the year at 365 days, or change it.
Read the chance of a match, the odds, and the table for other group sizes.

How it works

For n people and d days, the chance that all birthdays are different is (d / d) × ((d − 1) / d) × … × ((d − n + 1) / d). The chance of a shared birthday is 1 minus this product. If n is larger than d, the product is 0 and a match is certain.

Worked example

For 23 people in a 365-day year the product is about 0.4927, so the chance of a shared birthday is about 50.73%. With 22 people it is about 47.57%. For 57 people it first passes 99%.

Tips

The chance counts any pair, not a match with you. For a match with one given date, the numbers are much smaller.
Change the day count to model other coincidences, such as hash collisions.

Frequently asked questions

Why is it called a paradox?

It is not a contradiction, only surprising. With 23 people the chance of a shared birthday is already above 50%, far fewer than most people guess.

What is the formula?

The chance that everyone differs is the product of (365 − i) / 365 for i from 0 to n − 1. The chance of a match is 1 minus that. This tool multiplies the terms exactly.

When is a shared birthday certain?

With 366 people in a 365-day year, by the pigeonhole principle. Earlier the chance is just very close to 100%; the tool shows 100% once rounding reaches it.

What does this assume?

That birthdays are independent and every day is equally likely, and it ignores 29 February. Real birthdays are slightly uneven, which makes a match a little more likely.

More in Math

All tools in this category: Math ·Updated November 3, 2026