What is the probability that the second person has a birthday different from the first person?

Alright - we want to start out with that there is a 100% chance of someone having a birthday on any day. Next, we want to find the probability of someone happening to have the same birthday as them. Assuming that there are 365 days in the year (leap years!), there are 365 days to choose from. However, the amount of days we can choose is one. In the formula

(number of possibilities to choose) / (number of possibilities to choose from), we have 1 possibility, and 365 possibilities to choose from, making the probability 1/365