Consider $n$ people in a room.
On an arbitrarily chosen day $j$, what is the probability no one was born?
$\textbf{My Question}$: Is my thought process correct?
Assuming the probability of any individual born on any given day is independent, consider two individuals, James and Sarah.
$P$(No one was born on day $j$)=$P$("James not born on day $j$")$P$("Sarah not born on day $j$").
On day $j$, the probability of a person being born is $\frac{1}{365}$.
Thus, $P$(No one was born on day $j$)=$(1-\frac{1}{365})$$(1-\frac{1}{365})$=$(\frac{364}{365})^2$.
For $n$, $P$(No one was born on day $j$)=$(\frac{364}{365})^n$.
Reference:
Blitzstein, J. K., & Hwang, J. (2014). $\textit{Introduction to probability.}$
Best Answer
Absolutely correct.
A more advanced question would be "What's the minimum number of people in the room for the probability of two of them having the same birthday to be larger than 50%?". You will find an interesting answer. Have fun!