The Problem

In a room full of $r$ people, what are the chances that two people share the same birthday, assuming all birthdays are equally likely.

Equation $$ f(r)=1-\frac{365P_r}{365^r} $$ Breaking down the equation

• Things to note:
  • $365 P_r$ denotes all the permutations of birthdays with no duplication
  • $365^r$ denotes the number of permutations with duplicates allowed
    • therefore, to find the probability that there are no duplicates is calculated by calculating the ratio of permutations over perms with duplicates.
• therefore, to find the probability that there is at least one duplicate is the inverse of this probability.