Birthday Paradox Calculator

The Birthday Problem asks: how many people need to be in a room before there is at least a 50% chance that two of them share the same birthday? The answer — just 23 people — surprises most people, which is why it's also called the Birthday Paradox. It's a classic result in probability theory that highlights how our intuition about coincidences can be misleading.

With 23 people, there are 253 possible pairs of individuals (23 × 22 / 2). Even though each pair has only a 1/365 chance of sharing a birthday, the large number of pairs makes it more likely than not that at least one pair matches. The probability reaches approximately 50.7% with 23 people.

The probability that all N people have unique birthdays is P̄(D, N) = D × (D−1) × (D−2) × … × (D−N+1) / D^N. The probability that at least two share a birthday is simply 1 minus that value. This calculator computes both results for any combination of group size N and possible days D.

D represents the total number of equally likely outcomes — typically 365 for standard birthdays (ignoring leap years). You can change D to model other scenarios, such as 366 for leap years, or even abstract problems like hash collisions, where D might be millions.

With the standard 365 days, you need just 57 people to reach a 99% probability that at least two share a birthday. This calculator displays that threshold automatically so you can see how quickly the probability climbs with group size.

Yes, like most birthday paradox calculators, this tool assumes each day is equally likely as a birthday. In reality, birth rates vary slightly by day and season, but the uniform assumption is standard and produces a very good approximation for the true probability.

Absolutely. The underlying math applies to any problem where you draw N samples uniformly at random from D possibilities and want to know the probability of a collision. Common applications include hash table collisions in computer science, lottery coincidences, and DNA matching scenarios.

When N exceeds D, a shared value is guaranteed by the pigeonhole principle — the probability of a match is exactly 100%. For example, if there are only 365 possible birthdays and you have 366 people, at least two must share a birthday with certainty.