# 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.

## C++

`// C++ program to approximate number of people in Birthday Paradox ` `// problem ` `#include <cmath> ` `#include <iostream> ` `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); ` `} ` |

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## 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. ` |

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## 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". ` |

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## 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. ` |

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## PHP

`<?php ` `// PHP program to approximate ` `// number of people in Birthday ` `// Paradox problem ` ` ` `// Returns approximate number ` `// of people for a given probability ` `function` `find( ` `$p` `) ` `{ ` ` ` `return` `ceil` `(sqrt(2 * 365 * ` ` ` `log(1 / (1 - ` `$p` `)))); ` `} ` ` ` `// Driver Code ` `echo` `find(0.70); ` ` ` `// This code is contributed by aj_36 ` `?> ` |

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**Output :**

30

**Source:**

http://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

This article is contributed by **Shubham**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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