Why is 23 people enough for a 50% chance?

With 23 people there are 253 possible pairs. Each pair has a 1/365 chance of matching, and the combined probability across all pairs pushes past 50%. The math compounds quickly because you count every pair, not just pairs involving one fixed person.

What assumptions does the formula make?

The formula assumes 365 equally likely birthdays (leap days excluded) and that each person's birthday is independent of the others.

Does the calculator handle more than 365 people?

With 366 or more people the probability is exactly 100% by the pigeonhole principle — there are more people than possible birthdays, so at least one pair must share a date.

How is this related to cryptographic hash collisions?

Hash functions behave like a birthday problem: two different inputs that produce the same hash (a collision) become increasingly likely well before you exhaust the output space, just as shared birthdays appear long before the group fills all 365 slots.

Birthday Paradox Calculator

Find the probability that at least two people in a group share a birthday.

Group Size

Number of people

P(shared birthday)

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Probability vs. Group Size

Probability of at least two people sharing a birthday for groups of 1–100 people. The dashed line marks 50%.

50.7%

≥ 50% chance

99.0%

≥ 99% chance

99.9%

≥ 99.9% chance

Summary

The birthday paradox shows that in a group of just 23 people there is already a 50% chance two share a birthday — far higher than most expect. This calculator computes the exact probability for any group size using the standard combinatorial formula.

How it works

Enter the number of people in the group.
The calculator multiplies (365/365) × (364/365) × … × ((365−n+1)/365) to get the probability that all birthdays are distinct.
It then subtracts from 1 to give P(at least one shared birthday).
Results update instantly as you type.
The chart shows how probability grows from 1 to 100 people.

Use cases

Demonstrate counterintuitive probability in classrooms.
Estimate the likelihood of a birthday coincidence at a party or event.
Illustrate collision probability concepts in hash-function security.
Explore how quickly shared outcomes accumulate in random samples.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu