Birthday Paradox Calculator

The birthday paradox is one of the most quoted results in probability, and this calculator lets you test it for any group size. Enter how many people are in the room and it works out the chance that at least two of them share the same birthday. The surprise is how quickly that chance climbs: with only 23 people the odds are already better than even, at roughly 50.7 percent, and by 70 people it is a near certainty. It feels wrong because we instinctively compare our own birthday against everyone else, when the real question is whether any pair in the group matches, and the number of possible pairs grows very fast as the group gets larger. The calculator finds the answer the clean way, by first working out the probability that everybody has a different birthday and then subtracting that from one. It assumes 365 equally likely birthdays, treats each person independently, and ignores leap years and twins, which is the standard textbook model. Alongside the shared-birthday probability you also see the chance of no match at all and the number of distinct pairs in the group, which is the quantity really driving the result. Use it to settle an argument, to build intuition for how probability compounds, or to check a classroom example. Enter a group size and the answer updates straight away.

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people
50.7%
chance at least two people share a birthday
Chance of no match49.3%
People in group23
Possible pairs253

Based on 365 equally likely birthdays with each person independent. Leap years and twins are ignored, so the real-world figure is marginally higher. Estimate only.

How it works

The direct route, adding up every way a match could happen, is messy, so the calculator uses the complement. It works out the probability that all birthdays are different, then subtracts that from one. For a group of n people, the first person can have any birthday, the second must avoid one date so has a 364 in 365 chance of being different, the third has 363 in 365, and so on. Multiplying those fractions together gives the chance of no shared birthday: 365/365 times 364/365 times 363/365, down to (365 minus n plus 1)/365. One minus that product is the chance of at least one shared birthday. The number of pairs, n times (n minus 1) divided by two, is what makes the probability rise so fast, because every extra person adds a comparison against everyone already there.

Worked example

Take a group of 23 people. The chance that all 23 birthdays are different is 365/365 times 364/365 all the way down to 343/365, which comes to about 49.3 percent. Subtracting that from 100 percent leaves about 50.7 percent for at least one shared birthday, so a match is slightly more likely than not. Those 23 people form 253 distinct pairs, and it only takes one of those pairs to match. That is why a group barely larger than a sports team already tips past even odds, which matches the defaults shown in the calculator above.

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Frequently asked questions

Why is it called the birthday paradox?

It is called a paradox because the answer feels far too high. With just 23 people in a room the chance that at least two of them share a birthday is already about 50.7 percent, and with 70 people it is over 99.9 percent. Most people expect the group to need to be much larger, which is what makes the result surprising rather than being a genuine contradiction.

How is the shared birthday probability calculated?

The calculator works out the chance that everyone has a different birthday, then subtracts that from 1. For a group of n people the chance of all different birthdays is 365/365 times 364/365 times 363/365 and so on down to (365 minus n plus 1)/365. For 23 people that product is about 0.4927, so the chance of a shared birthday is about 0.5073, or 50.7 percent.

Does the calculator allow for leap years or twins?

No. It uses the standard model of 365 equally likely birthdays and treats each person as independent, so it ignores 29 February, seasonal birth patterns and twins. Real birth dates are slightly clustered, which pushes the true probability a little higher than the model, but the difference is small for ordinary group sizes.