Birthday Paradox Calculator
This birthday paradox calculator shows that with just 23 people the chance that two share a birthday is already 50.7 %. It works for any group size and in reverse.
Source: Wikipedia – Geburtstagsparadoxon. Updated: .
How it is calculated
The birthday paradox is not a true contradiction but a surprising result: in a group of just 23 people it is more likely than not that two share a birthday. It feels impossible because we instinctively think of our own birthday.
How it is calculated
The opposite is easier: all birthdays different. The second person has 364 of 365 free days, the third 363, and so on. The probability that all are different is the product 365/365 · 364/365 · 363/365 · …, and the probability we want is 1 minus that product. For 23 people it gives 50.7 %, for 30 it is 70.6 %, for 50 already 97.0 % and for 70 people 99.9 %.
Why it is not really paradoxical
What counts is not one particular person but all pairs: 23 people form 253 pairs, each with a chance of 1/365. The probability that someone shares your birthday grows much more slowly: with 22 other people it is only 5.9 %. The effect underlies collisions in hash values (the birthday attack).
- With 366 days (leap year) the values change only slightly.
- With more people than days, a shared birthday is certain (pigeonhole principle).
- The table shows typical group sizes from 5 to 100 people.
Frequently asked questions
How many people do you need for 50 % in the birthday paradox?
23 people. That is the first group size where the probability of at least two shared birthdays, 50.7 %, exceeds one half.
How do you calculate the birthday paradox?
Multiply the probabilities that each further person has a still free day: 365/365 · 364/365 · …, and subtract the result from 1.
Why is the probability so high?
Because not one single birthday is compared but all pairs. 23 people make 253 pairs, 50 people already 1,225.
Sources and legal basis
As of: · Editorial policy
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