probability - What's the smallest $n$ so that it's more likely than not that someone among the $n$ people has the same birthday as you?

Suppose that the birthdays of different people in a group of $n$ people are independent, each equally likely to be on the $365$ possible days. (Pretend there's no such thing as a leap day.)

What's the smallest $n$ so that it's more likely than not that someone among the $n$ people has the same birthday as you? (You're not part of this group.)

At first, I get $n=23$ by using the complement rule to get the probability that someone has the same birthday as me $(P\{\text{same birthday}\} = 1 - P\{\text{different birthday}\})$ and use $e^{-x}=1-x$.

But I don't think the answer is $23$ since I am not part of the group.

Answer

HINT

The probability that none of the $n$ people have the same birthday as you is:

$$\big( \frac{364}{365} \big)^n$$

So, set this equal to $\frac{1}{2}$ and let WolframAlpha do its magic.

And as you alrady suspected, this problem is not like the infamous 'Birthday Problem'