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++ program to approximate number of people in Birthday Paradox
// Returns approximate number of people for a given probability
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