There are $n$ people present in a room. Assume that people’s birthdays are equally likely to be on any day of the year. What is the probability that at least one of them celebrates their birthday on the same day as you? How large does n need to be for this probability to be more than 1/2 ?
My approach was as follows: $$ P(\text{at least one shares my birthday}) = 1 - P(\text{no one shares my birthday})\\ = 1-\left(\frac{364}{365}\right)^n $$ This is greater than 1/2 if
$$ n \text{ log} \frac{364}{365} \leq \text{ log}\frac{1}{2} \implies n \geq 253 $$
I'd appreciate it if somebody could say whether they think this is correct or not.