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?
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.

## C++

 // C++ program to approximate number of people in Birthday Paradox   // problem  #include  #include  using namespace std;     // Returns approximate number of people for a given probability  int find(double p)  {      return ceil(sqrt(2*365*log(1/(1-p))));  }     int main()  {     cout << find(0.70);  }

## Java

 // Java program to approximate number  // of people in Birthday Paradox problem  class GFG {             // Returns approximate number of people       // for a given probability      static double find(double p) {                     return Math.ceil(Math.sqrt(2 *               365 * Math.log(1 / (1 - p))));      }             // Driver code      public static void main(String[] args)       {                     System.out.println(find(0.70));       }  }     // This code is contributed by Anant Agarwal.

## Python3

 # Python3 code to approximate number  # of people in Birthday Paradox problem  import math     # Returns approximate number of   # people for a given probability  def find( p ):      return math.ceil(math.sqrt(2 * 365 *                      math.log(1/(1-p))));     # Driver Code  print(find(0.70))     # This code is contributed by "Sharad_Bhardwaj".

## C#

 // C# program to approximate number  // of people in Birthday Paradox problem.  using System;     class GFG {              // Returns approximate number of people       // for a given probability      static double find(double p) {                      return Math.Ceiling(Math.Sqrt(2 *               365 * Math.Log(1 / (1 - p))));      }              // Driver code      public static void Main()       {               Console.Write(find(0.70));       }  }      // This code is contributed by nitin mittal.

## PHP

 

Output :

30

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

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Improved By : nitin mittal, jit_t

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