 However, it is simpler to calculate , the probability that no two people in the room have the same birthday.Then, because and are the only two possibilities and are also mutually exclusive, 

In deference to widely published solutions concluding that 23 is the minimum number of people necessary to have a that is greater than 50%, the following calculation of will use 23 people as an example.If one numbers the 23 people from 1 to 23, the event that all 23 people have different birthdays is the same as the event that person 2 does not have the same birthday as person 1, and that person 3 does not have the same birthday as either person 1 or person 2, and so on, and finally that person 23 does not have the same birthday as any of persons 1 through 22.