# IF THERE ARE 'N' NUMBER OF PEOPLE WHAT IS THE PROBABILITY THAT TWO PEOPLE WILL HAVE BIRTHDAY ON SAME DAY

justaguide | Certified Educator

This problem has been solved with an assumption that all years have 365 days, no leap years.

Instead of calculating the probability of at least 2 people in the group of N people having their birthday on the same day, we can calculate the probability that none of them have a birthday on the same day and then subtract that from 1.

Let's start with 2 people. The probability that they don't have a common birthday is (365/365)*(364/365). This gives the probability that they have the birthday on the same day as 1 - (365/365)*(364/365) = 1 - (365*364)/(365)^2

Next take 3 people. The probability that at least 2 share a common birthday is 1-(365*364*363)/(365)^3

Continuing, for N people the probability that at least have the birthday on the same day is:

1 - (365*363*363*...*(365 - n +1))/365^n

=> 1 - 365!/ [(365 - n)!*365^n]

The required probability that two people in a group of N have the birthday on the same day is 1 - 365!/ [(365 - n)!*365^n]