What is the probability that on any chosen given day (e.g. today) there is at least one person (in a group of N) who is celebrating his birthday?
I would say the answer is either $N/365$ because you get $\frac{1}{365}+\frac{1}{365}$..$N$ times or $1-(\frac{364}{365})^N$ because this is $1$ - the chances of all N people having birthday on a different day other than the chosen one.
Which is the right one(if any is) and why?
Your second answer is right. What if there were more than 365 people in the group? Then your first answer would produce a number larger than 1.
You can only add probabilities for "or" when the events are disjoint. In this case, it is possible that more than one person has their birthday on that day, so you can't just use the addition rule without subtracting out some overlap.
The last one i.e $1-(\frac{364}{365})^N$ is the correct answer[Here $\frac{364}{365})^N$ is the probability that $N$ people has birthday in all the days except your specified day & you take the complement of that event, which is needed.] . Because your 1st answer implies choosing $N$ days among $365$ days in a year. Does satisfy your requirment.
