Probability of 2 person has the same birthday in a class.

birthdayprobability

problem: there are n persons in a room, what is the probability that no two of them celebrate the same birthday in a year?
Here is my thought process,
The sample space is $|\{(b_1,b_2,\dots,b_n): b_1,b_2\in\{1,2,\dots 365\}\}|=(365)^n$ , and I got stuck at counting the Event, $|\{(b_1,b_2,\dots,b_n)：b_i\neq b_j \forall i\neq j \}|=365*364*\dots*1$ but what if n>365? how do I count that?

Best Answer

  • Your first calculation is $365$,

  • your second calculation $365\times 364$,

  • ...,

  • your $365$th calculation $365\times 364\times \cdots\times 1$,

  • and your $366$th calculation $365\times 364\times \cdots\times 1\times 0$ which is $0$.

You can carry on further, but you will always have the $\times 0$ term with more people. So whenever you have more people than possible birthdays, you never get them each having a different birthday.

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