Brain Teasers
Birthdays
Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.Probability
How many people do you need to enter a room before the probability of any 2 or more people sharing a birthday (day and month only, not year) is greater than 50%?
Assume for the sake of the puzzle that birthdays in the population at large are equally spread over a 365 day year.
Assume for the sake of the puzzle that birthdays in the population at large are equally spread over a 365 day year.
Hint
It is easier to calculate the probability that all birthdays in the group are unique.Answer
We get the solution by calculating the solution to the opposite case: How many people are necessary before the chance of them all having unique birthdays is less than 50%.Let Pn be the probability of n people having unique birthdays.
Obviously, with 1 person in the room, P1 = 1.0
If there are n people in the room, the probability of the (n+1)th person also having a unique birthday is (365-n)/365.
Hence P(n+1) given Pn is (365-n)/365.
So P(n+1) = Pn * (365-n)/365.
If we start with n=1 and count upwards, calculating Pn, we find that P22 = 0.52 and P23 = 0.49.
Hence, with 23 people in the room, the probability that 2 or more people will share a birthday is 51%.
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