Birthday Paradox Calculator

Enter the number of people in the group.

Enter the number of equally likely possible birthdays. Use 365 for the classic birthday problem or 366 when deliberately modelling leap day as an equally likely outcome.

The result shows the probability of at least one shared birthday, the probability that every birthday is different, the number of possible pairs, and common probability thresholds.

The birthday paradox feels surprising because people often compare everyone only with themselves.

The actual problem compares every possible pair in the group.

A group of 23 people contains 23 × 22 ÷ 2, or 253, distinct pairs.

Under the classic 365-day model, those comparisons produce a 50.7297% probability that at least one pair shares a birthday.

It is easier to calculate the probability that every birthday is different and then subtract that value from one.

The first person can have any birthday. The second must avoid one occupied day, the third must avoid two occupied days, and the process continues through the group.

For n people and d equally likely possible birthdays, the no-match probability is the product of (d − i) ÷ d for i from zero through n − 1.

When the group contains more people than possible birthdays, a match is guaranteed by the pigeonhole principle.

The classic problem asks whether any pair in the group shares a birthday.

That is very different from asking whether at least one other person shares one specific person's birthday.

With 365 equally likely birthdays, 23 total people give a 50.7297% chance that some pair matches.

A specific person needs 253 other people, or 254 people including that person, before the chance that someone matches that particular birthday exceeds 50%.

A group of n people contains n(n − 1)/2 distinct pairs.

Under a uniform d-outcome model, each pair has probability 1/d of matching.

The expected number of matching pairs is therefore n(n − 1)/(2d).

Expected matching pairs is not the same as the probability of at least one match. It can exceed one because a group can contain several matching pairs.

For 365 equally likely birthdays, 23 people are required to exceed 50%, 41 people exceed 90%, and 57 people exceed 99%.

The live calculator recomputes these minimum group sizes when the number of possible outcomes changes.

The same mathematics applies whenever independent items are assigned uniformly to a fixed number of possible outcomes.

Examples include simplified hash collisions, randomly assigned identifiers, numbered bins, repeated random selections, and other occupancy problems.

The model is not valid when outcomes have different probabilities unless the underlying calculation is changed.

The standard birthday problem assumes 365 equally likely birthdays, independent people, no leap day, and no special relationship among births.

These assumptions make the probability exactly calculable and useful for teaching collision behaviour.

Real populations have seasonal patterns, scheduled deliveries, twins, leap-day births, and other departures from the model.

There is no logical contradiction.

It is called a paradox because the correct probability conflicts with many people's first intuition.

The intuition improves once the rapidly growing number of pair comparisons is made visible.

The calculator sums logarithms instead of directly multiplying many probabilities close to one.

It uses a stable exponential-complement calculation so very small collision probabilities are not unnecessarily rounded to zero during calculation.

Displayed probabilities use up to four decimal places, while calculations retain browser floating-point precision.

The calculation runs entirely in the browser.

Group sizes, possible outcomes, and results are not uploaded, added to a public URL, or permanently stored.