Birthday Paradox Calculator
Online calculator for calculating the birthday paradox
This function calculates the Birthday Paradox for a set of n people.
The birthday paradox descript the probability that, in a set of n people, at least two will share a birthday. The birthday paradox refers to the counterintuitive fact that only 23 people are needed for that probability to exceed 50%.
For the calculation, enter the number of people. Then click on the 'Calculate' button.
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The red curve in the image shows the probability that at least two people have the same birthday. The blue curve shows the probability that all people have birthdays on different days.
Formula
\(\displaystyle \overline{p}(n)=1 - \left(1-\frac{1}{365}\right)×\left(1-\frac{2}{365}\right)× ...×\left(1-\frac{n-1}{365}\right)\)
Example
The following example shows the calculation for 3 people.
\(\displaystyle 1 - \left(1-\frac{1}{365}\right)×\left(1-\frac{2}{365}\right)=0.0082 = 0.82\%\)
The following example shows the calculation for 5 people.
\(\displaystyle 1 - \left(1-\frac{1}{365}\right)×\left(1-\frac{2}{365}\right) ×\left(1-\frac{3}{365}\right)×\left(1-\frac{4}{365}\right)=0.0271 = 2.71\%\)
Description
When asked what is the probability that out of 23 people at least two of them have their birthday on the same day of the year, the answer is astonishing to most people and is therefore perceived as paradoxical.
Most people misjudge the probability by a power of ten. It is not (as is usually estimated) between 1% and 5%, but over 50%, and for 50 people even over 97%.
In contrast, for a probability of 50% that someone's birthday is on a specific day, 253 people are necessary. The reason for this hugh difference is that with each person added, the number of possible couples who share a birthday on the same day increases.
As the number of couples increases more rapidly as the number of people grows, the probability that two people in the group have the same birthday also increases. Therefore, 253 people are probably necessary to cover 50% of the different data.
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