How would i calculate probability of something with multiple variables?

Question

I wanted to figure out percentage chance of it being a persons birthday within a group of people.

Answer 1

1 - (364/365)^N, where N is the number of people in the room.

Answer 2

For simplicity, let's ignore the existence of leap years and assume every year is 365 days long. I'll start with a "group" of one person and then increase the group size so that you can see the pattern. Furthermore, it is sometimes easier to calculate the probability that some event did not happen and then subtract from 1 to find the probability that the event did happen; I will use that approach here.

If there is only one person, the probability it's not their birthday is 364/365 (the probability it is their birthday is 1 - (364/365) = 1/365).

If there are two people, there is a 364/365 probability that it's not the first person's birthday and a 364/365 probability that it's not the second person's birthday. So the probability it is one of their birthdays is 1-(364/365)², which is very slightly less than 2/365 because the two people could have the same birthday.

If there are three people, the probability it isn't anyone's birthday is then (364/365)³, so the probability it is someone's birthday is 1-(364/365)³.

Hopefully you can see the pattern now and for a group of size n, the probability it is someone's birthday is 1-(364/365)ⁿ.

This problem is related to the somewhat tricker problem known as the Birthday Paradox: https://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox/.