A group of 30 people is selected at random. What is the probability that at least 2 of them will have the same birthday?

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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A group of 30 people is selected at random. What is the probability that at least 2 of them will have the same birthday?

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Given : A group of 30 people is selected at random.  

Question : What is the probability that at least 2 of them will have the same birthday? 

Probability ( at least 2 people share the same birthday ) is also can be written as :

Probability ( at least 2 people share the same birthday ) =  1 - P( no two people share the same birthday) 

First compute the probability that two people share different birthday . 

Let us say that , first person we pick has a birthday and it can be on any day of the year i.e. any of the 365 days . 

So it will be : 365365 

The next person, person 2, can have a birthday on any day of the year except for the one that person 1 has birthday on a particular day , 

So it will be : 365365

Simultaneously , two of them having different birthdays is :

365365 x 365365

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