Birthday Paradox

How many people must be there in a room to make the probability 100% that at-least two people in the room have same birthday? 
Answer: 367 (since there are 366 possible birthdays, including February 29). 
The above question was simple. Try the below question yourself.

How many people must be there in a room to make the probability 50% that at-least two people in the room have same birthday? 
Answer: 23 
The number is surprisingly very low. In fact, we need only 70 people to make the probability 99.9 %.
Let us discuss the generalized formula.

What is the probability that two persons among n have same birthday? 
Let the probability that two people in a room with n have same birthday be P(same). P(Same) can be easily evaluated in terms of P(different) where P(different) is the probability that all of them have different birthday.
P(same) = 1 - P(different)
P(different) can be written as 1 x (364/365) x (363/365) x (362/365) x .... x (1 - (n-1)/365)
How did we get the above expression? 
Persons from first to last can get birthdays in following order for all birthdays to be distinct: 
The first person can have any birthday among 365 
The second person should have a birthday which is not same as first person 
The third person should have a birthday which is not same as first two persons. 
................ 
............... 
The n'th person should have a birthday which is not same as any of the earlier considered (n-1) persons.

Approximation of above expression 
The above expression can be approximated using Taylor's Series. 

provides a first-order approximation for ex for x << 1:

To apply this approximation to the first expression derived for p(different), set x = -a / 365. Thus,

The above expression derived for p(different) can be written as 
1 x (1 - 1/365) x (1 - 2/365) x (1 - 3/365) x …. x (1 – (n-1)/365)
By putting the value of 1 - a/365 as e^-a/365, we get following.



Therefore,
p(same) = 1- p(different) 

An even coarser approximation is given by
p(same) 

By taking Log on both sides, we get the reverse formula.

Using the above approximate formula, we can approximate number of people for a given probability. For example the following C++ function find() returns the smallest n for which the probability is greater than the given p.

Implementation of approximate formula. 

The following is program to approximate number of people for a given probability.  

30

Time Complexity: O(log n)
Auxiliary Space: O(1)

Source: 
https://en.wikipedia.org/wiki/Birthday_problem

Applications: 
1) Birthday Paradox is generally discussed with hashing to show importance of collision handling even for a small set of keys. 
2) Birthday Attack
Below is an implementation:

Probability to find : 
Total no. of people out of which there is 0.0 probability that two of them have same birthdays is 239 

Time Complexity: O(log n)
Auxiliary Space: O(1)