The answer is 23! When there are 23 people in a room together there is greater than a 50% (50.7297%) chance that two individuals share a birthday.
To find this solution we want to use the compliment. This means that we will find the probability that nobody shares a birthday, and subtract that value from one for our solution.
The probability that two individuals do not share a birthday is (365/365) x (364/365) = .9973
1-.9973=.0027
The Probability three individuals do not share a birthday is (365/365) x (364/365) x (363/365) = .9917
1-.9917=.0082
Doing this helps us find an emerging pattern that allows us to generalize our formula in terms of n, where n is the number of people in the room.
1- [(365!/(365-n)!/365^2)]
When plugging in values for n you will quickly see that when n=23, this is the first time the probability reaches 50%. The table below shows the probability for many different values allowing a quick and easy way for you to check the probabilities for some other scenarios!
To find this solution we want to use the compliment. This means that we will find the probability that nobody shares a birthday, and subtract that value from one for our solution.
The probability that two individuals do not share a birthday is (365/365) x (364/365) = .9973
1-.9973=.0027
The Probability three individuals do not share a birthday is (365/365) x (364/365) x (363/365) = .9917
1-.9917=.0082
Doing this helps us find an emerging pattern that allows us to generalize our formula in terms of n, where n is the number of people in the room.
1- [(365!/(365-n)!/365^2)]
When plugging in values for n you will quickly see that when n=23, this is the first time the probability reaches 50%. The table below shows the probability for many different values allowing a quick and easy way for you to check the probabilities for some other scenarios!