How many people must be there in a room to make the probability 100% that 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 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.

C++ Implementation of approximate formula.
The following is C++ program to approximate number of people for a given probability.

`// C++ program to approximate number of people in <a href="#">Birthday Paradox</a> `
`// 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);`
`}`

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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source and credits.