Birthday Paradox Calculator
Discover the surprisingly high probability that at least two people in a group share a birthday. Use it to demonstrate the birthday paradox in classrooms or trivia nights.
About this calculator
The birthday paradox calculates the probability that at least two people in a group of n share the same birthday out of d possible days (typically 365). Rather than computing shared birthdays directly, we use the complement: P(at least one shared) = 1 − P(all birthdays unique). The probability that all n birthdays are different is: P(unique) = (d/d) × ((d−1)/d) × ((d−2)/d) × … × ((d−n+1)/d). Each term represents the probability that each successive person has a birthday not yet taken. Subtracting from 1 gives the paradox probability. Remarkably, with just 23 people the probability exceeds 50%, and with 70 people it surpasses 99.9%.
How to use
Try a group of 23 people in a standard 365-day year. Enter people = 23 and days = 365. The calculator multiplies: (365/365) × (364/365) × … × (343/365), producing P(all unique) ≈ 0.4927. Subtracting: 1 − 0.4927 = 0.5073, or about 50.7%. That means in a room of just 23 strangers, it is more likely than not that two of them share a birthday — a result most people find astonishing.
Frequently asked questions
Why does the birthday paradox probability reach 50% with only 23 people?
The intuition breaks down because we are not asking if someone shares YOUR birthday — we are asking if any two people among all possible pairs share a birthday. With 23 people there are C(23,2) = 253 distinct pairs, each with a 1/365 chance of matching. While no single pair is likely to match, 253 simultaneous chances make a shared birthday more probable than not. This combinatorial explosion is what makes the result feel paradoxical.
What happens to the birthday paradox probability if you use a leap year with 366 days?
Using 366 days instead of 365 shifts the curve very slightly to the right — you now need about 23 or 24 people to cross the 50% threshold rather than exactly 23. The effect is minimal because the difference between 365 and 366 possible birthdays is less than 0.3%. For practical purposes, whether you use 365 or 366 days, the paradox result is essentially the same.
How many people do you need in a group for a 99% birthday match probability?
You need approximately 57 people for the probability of a shared birthday to exceed 99% in a 365-day year. By 70 people the probability is already 99.9%. The growth is rapid: 10 people gives about 11.7%, 30 people gives 70.6%, and 50 people gives 97%. This steep rise illustrates why the birthday paradox is a favourite example in probability courses — the threshold is much lower than human intuition suggests.