Enter the number of people in a group and the number of days in a year to calculate the Birthday Paradox probability. You'll see the probability that at least two people share a birthday, the probability that all birthdays are unique, and how your group size compares to the famous 50% threshold of 23 people.
Probability of a Shared Birthday
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Probability All Birthdays Are Unique
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People Needed for 50% Chance
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People Needed for 99% Chance
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The Birthday Paradox is a famous probability problem that asks: how many people need to be in a room for there to be a 50% chance that at least two of them share the same birthday? The surprising answer is just 23 people — far fewer than most people intuitively expect, which is why it's often called a paradox.
The probability that all N people have unique birthdays in a year of D days is P_unique = (D/D) × ((D-1)/D) × ((D-2)/D) × ... × ((D-N+1)/D). The probability of at least one shared birthday is then P_shared = 1 - P_unique. As N grows, P_shared rises surprisingly fast toward 100%.
With 23 people and a standard 365-day year, the probability that at least two people share a birthday is approximately 50.7%. This is the famous threshold at the heart of the Birthday Paradox — most people expect you'd need far more than 23 people to reach 50%.
With 100 people in a group, the probability that at least two share a birthday is approximately 99.9999% — virtually certain. By the time you reach about 57 people, the probability already exceeds 99%.
It's not a true logical paradox — there's no contradiction in the math. It's called a paradox because the result is deeply counterintuitive. Most people assume you'd need around 183 people (half of 365) for a 50% chance of a shared birthday, but the correct answer of 23 feels impossibly low. It's better described as a veridical paradox: a surprising truth.
Our intuition tends to compare one specific person's birthday against everyone else's. But the birthday problem counts every possible pair in the group. With 23 people, there are 253 unique pairs — each with its own chance of a match. The cumulative probability across all those pairs quickly adds up, producing a result that surprises almost everyone.
You need 70 people for a greater than 99.9% probability of at least one shared birthday, and just 57 people to exceed 99%. With 366 people, you are guaranteed a shared birthday by the pigeonhole principle (more people than days).
Yes — the birthday problem underlies a cryptographic attack called the 'birthday attack,' used to find collisions in hash functions. It also appears in fields like statistics, data science, and information security to model the likelihood of collisions when sampling from a finite set.