Birthday Paradox
Calculate the probability that at least two people share a birthday in any group size. Free online birthday paradox probability calculator with interactive charts and breakdowns.
About This Calculator
The Birthday Paradox Calculator computes the probability that at least two people in a group share a birthday. This famous probability problem reveals counterintuitive results — in a group of just 23 people, there is a 50% chance of a shared birthday, and with 57 people, the probability exceeds 99%.
The calculation uses the complementary probability approach. First, it computes the probability that no two people share a birthday: for the first person, all 365 days are available; for the second, only 364; for the third, 363; and so on. The product of these fractions gives the probability of no match. Subtracting from 1 gives the probability of at least one shared birthday.
The calculator also shows the number of unique pairs (n(n-1)/2), the expected number of shared birthdays, and the minimum group sizes needed for 50% and 99% probability thresholds. The probability curve chart visualizes how the likelihood grows with group size.
Regional Notes
Global: The birthday paradox is universal and applies to any population regardless of geographic region. The formula uses 365 days by default, but you can adjust for leap years (365.25 days) or any other number of days.
IN / US / UK: The same mathematics applies worldwide. Birthday distributions are approximately uniform across all countries, making this a truly universal probability tool.
Frequently Asked Questions
What is the birthday paradox?
The birthday paradox is the counterintuitive probability phenomenon where in a group of just 23 people, there is a 50% chance that at least two people share a birthday. The probability rises to 99% with just 57 people.
Is the birthday paradox actually a paradox?
No, the birthday problem is not a true logical paradox. It is called a paradox because the results seem counterintuitive — most people guess a much higher number is needed. It is a veridical paradox: a result that seems absurd but is mathematically true.
How do you calculate the birthday paradox probability?
The probability of at least one shared birthday is 1 minus the probability of no shared birthdays. For n people, the probability of no match is (365/365) × (364/365) × (363/365) × ... × ((365-n+1)/365). Subtract this from 1 to get the probability of a shared birthday.
What is the minimum group size for a 50% chance of shared birthday?
With 365 days in a year, only 23 people are needed for a 50% chance of at least one shared birthday. This is the most famous result of the birthday problem.
What group size guarantees a shared birthday?
With 365 days, you need 366 people to guarantee (100% probability) that at least two share a birthday by the pigeonhole principle. With 57 people, the probability already reaches 99%.
Does the birthday paradox apply to any number of days in a year?
Yes, the birthday paradox formula works for any number of days. For example, with 365.25 days (accounting for leap years), you need slightly more people: about 24 for 50% chance and 58 for 99% chance.
Where is the birthday paradox used in real-world applications?
The birthday paradox is used in cryptography (birthday attacks on hash functions), DNA matching, lottery analysis, and any application involving collision probability in large datasets.
How many pairs of people are in a group of n people?
The number of possible pairs in a group of n people is n(n-1)/2. For n=23, there are 253 pairs — this is why the probability rises so quickly despite there being 365 days.