Inspired by the Birthday Problem

Question

Let

${n}$ = Number of people in my friends list = 336 people;

${X}$ = Event no one shares the same birthday in $n$.

${Y}$ = Event everyone has distinct birthdays in $n$

$$ P(X) + P(Y) = 1$$

$$ P(Y) = \frac{365!}{(365-n)!(365)^n}$$

I understand the birthday problem, but the following question was driving me crazy:

My question is, what is the probability that a person- James, has a distinct birthday from the others.

Answer 1

$X=\text{James shares birthday with atleast one more person}$

$Y=\text{All other }n-1\text{ people have different birthday than James}$

Again, $P(X)+P(Y)=1$.

To find $P(Y)$ note that birthday of James can be any of $365$ days. Fixing birthday of James, birthdays for the other $n-1$ people can be chosen from any of the remaining $364$ days, in $364^{n-1}$ ways, giving total $365\times364^{n-1}$ ways for birthdays.

Without any restriction, there are $365^n$ ways the birthdays can be selected.

So $P(Y)=\frac{365\times364^{n-1}}{365^n} = \left(\frac{364}{365}\right)^{n-1}$, and hence $P(X)=1-\left(\frac{364}{365}\right)^{n-1}$.