Birthday Paradox Calculator
Calculate the probability that two people in a group of n share a birthday. Explore the surprising birthday paradox with a probability curve.
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Input
Enter the number of people to find the probability that at least two of them share a birthday.
Days in a year. The default is 365.
Result
Probability of a shared birthday among 23 people
50.73percent
Probability all are different
49.27 percent
People needed to pass 50 percent
23 people
Calculation steps
Treat the year as 365 days, then multiply the chance that each person avoids the earlier birthdays.
For 365 days, the probability that all 23 people differ is 49.27 percent.
Subtracting the all-distinct probability from 1 gives a match probability of 50.73 percent.
How it works
- The probability that everyone has a different birthday is the running product 365/365 times 364/365 and so on. Written as a formula it is 365! divided by ((365 minus n)! times 365 to the power n).
- The probability that at least two people share a birthday is 1 minus the all-distinct probability.
- With 23 people the matching probability is about 50.7 percent, just over half. A shared birthday becomes likely with a surprisingly small group, which is why this is called the birthday paradox.
- The number of days defaults to 365 (ignoring leap years) but you can change it. Changing the days also changes the group size needed to pass 50 percent.
Frequently asked questions
- How is the probability of a shared birthday calculated?
- First the probability that everyone's birthday differs is found by multiplying 365/365 × 364/365 × …, which equals 365! ÷ ((365 − n)! × 365ⁿ). The chance that at least two people match is 1 minus that value.
- Why does the probability reach about 50% at just 23 people?
- With 23 people the match probability is about 50.7%, already past half. Because the number of pairs being compared grows quickly as people are added, a match becomes likely with far fewer people than intuition suggests — that is the birthday paradox.
- Can I use a number of days other than 365?
- Yes. The tool defaults to 365 days (ignoring leap years), but you can change the number of days in the input field. Changing it also shifts the group size at which the probability crosses 50%.
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