Birthday Paradox Calculator
Calculate the probability that at least two people in a group share a birthday. Useful for teaching probability concepts or settling that classic party debate.
About this calculator
The birthday paradox is a famous result in probability theory showing that coincidences occur far more often than intuition suggests. The core calculation finds the probability that all n people have different birthdays: P(all different) = (365/365) × (364/365) × (363/365) × … × ((365 − n + 1)/365). More compactly: P(all different) = ∏ᵢ₌₀ⁿ⁻¹ (d − i) / d, where d is the number of days in a year. The complement gives the probability that at least two people share a birthday: P(at least one match) = 1 − P(all different). With only 23 people this probability already exceeds 50%, and with 70 people it surpasses 99.9%, which is why the result feels paradoxical.
How to use
Let's find the probability of a shared birthday in a group of 23 people using a standard 365-day year. Step 1 — Compute P(all different): multiply (365/365) × (364/365) × (363/365) × … × (343/365) for 23 terms. This product ≈ 0.4927. Step 2 — Compute P(at least two share): 1 − 0.4927 = 0.5073. So there is approximately a 50.73% chance that at least two people in a room of 23 share a birthday. Enter 23 as group size, 365 as days in year, and select 'at least two' to see this result.
Frequently asked questions
Why is the birthday paradox probability so much higher than most people expect?
Human intuition tends to anchor on the probability that someone shares YOUR specific birthday, which is indeed low for small groups. The paradox arises because we are asking whether any two people among all possible pairs share a birthday. With 23 people there are 253 unique pairs, each with a 1/365 chance of matching — these chances compound rapidly. This is a classic example of how humans underestimate combinatorial growth.
How many people do you need for a 99% chance of a shared birthday?
You need just 57 people to reach a 99% probability of at least one shared birthday. At 70 people the probability climbs to approximately 99.9%. This is startlingly low compared to the 365 people most people guess. The calculation follows the same complement method: keep multiplying (365 − i)/365 until the product drops below 0.01, then add 1 for the group size.
What happens to the birthday paradox probability if you use a leap year with 366 days?
Using 366 days instead of 365 slightly lowers the probability of a shared birthday for any given group size, because birthdays are spread over more possible dates. For example, with 23 people and 366 days, P(at least one match) drops from about 50.7% to approximately 50.6% — a negligible difference. The effect of extra days becomes more visible in very small groups but is always minor compared to the dominant role of group size.