Same birthdays
Task number: 3536
What is the probability that at least two of twenty people will have their birthdays on the same day?
Solution
We represent people's birthdays using a mapping from the given set of people to the days in the year.
We first consider that the probability that no two people in the group have the same birthday is given by the ratio of the number of injective mappings to the total number of mappings.
The probabilty we seek will be given by the difference between this ratio and one, so for \(n\) people and a 365-day year this equals \[ p(n)=1-\frac{365!}{(365-n)!\cdot 365^n}= 1-\frac{364{\cdot} 363\cdots(366-n)}{365^{n-1}}. \]
Answer
The probability approximately equals \(p(20)\doteq 42\%\).
