Birthday Paradox Calculator

Explore birthday collision probability, no-match probability, expected matching pairs, and common probability thresholds.

Group and calendar

Calculate the chance that at least two people share a birthday.

People in the group

Possible birthdays

At least one shared birthday

50.729723%

No shared birthday

49.270277%

Expected matching pairs

0.6932

50% probability reached at

23 people

90% probability reached at

41 people

99% probability reached at

57 people

Model assumptions

The classic model treats birthdays as independent and uniformly distributed, excludes leap day, and asks whether any pair matches. Real birth dates are not perfectly uniform.

Why the result feels surprising

The comparison is not one person's birthday against everyone else. Every pair in the group can match, so the number of comparisons grows quickly.

With 23 people and 365 equally likely birthdays, the probability of at least one shared birthday is just over 50%.

Model boundaries

The classic calculation assumes independent, uniformly distributed birthdays and normally uses 365 days. Seasonal birth patterns and leap day make real populations slightly different.

Formula variables

$n$: Number of people
$d$: Number of equally likely birthdays

No shared birthday

P(no match)=\prod_{i=0}^{n-1}\frac{d-i}{d}

At least one match

P(match)=1-P(no match)

Expected matching pairs

E=\frac{n(n-1)}{2d}

Examples

Classic 23-person group

Input

23 people; 365 possible birthdays

Show result

Result

About 50.73% chance of at least one shared birthday

Frequently Asked Questions

Does this mean someone matches my birthday?

No. It asks whether any pair in the entire group shares a birthday.

Why can I change the number of days?

It lets the same collision model represent other equally likely outcome spaces.

References

NIST/SEMATECH e-Handbook — Probability Distributions

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