Birthday Paradox
Formula and examples for calculating the probability of shared birthdays
Overview
The birthday paradox describes the probability that in a group of n people, at least two people share the same birthday.
The result is counterintuitive: with only 23 people, the probability
already exceeds 50%.
The complement approach is used: first calculate the probability that all birthdays are different,
then subtract from 1.
Formula and Examples
General formula
\(\displaystyle p(n)=1-\prod_{k=1}^{n-1}\left(1-\frac{k}{365}\right)\)
Example for 3 people
\(\displaystyle p(3)=1-\left(1-\frac{1}{365}\right)\left(1-\frac{2}{365}\right)=0.82\%\)
Example for 5 people
\(\displaystyle p(5)=1-\prod_{k=1}^{4}\left(1-\frac{k}{365}\right)=2.71\%\)
\(\displaystyle p(n)=1-\prod_{k=1}^{n-1}\left(1-\frac{k}{365}\right)\)
- Probability grows quickly with group size
- At 23 people: probability is above 50%
- At 50 people: probability is already above 97%
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