Solution: Happy Birthday Math Puzzle

Here are our answers.

One way to solve this is to turn the problem around and think about how likely it is for there to be NO matches in a group of a given size. If there is only one person in a room, there can be no shared birthdays, since there is no one to share with. The probability of not having a match in this case is 1. Events that are certain are said to have a probability of 1. At the other extreme, with 367 people in the room, it is certain that there will be at least one shared birthday, since there aren’t enough birthdays to go around for that many people.

Now imagine that a second person walks into the room. The probability of that person not having the same birthday as the first occupant of the room is 365 / 366, or 0.997. There are 366 possible birthdays, and only one of them is a match.

Now if the first two people in the room have different birthdays and a third person walks in, there are two days used up, so the probability of the third person not sharing a birthday with either roommate is 364 / 366 and the probability of no sharing among the three of them is 1 x 365 / 366 x 364 / 366 = 0.992, which is still over 99%. So with two or three people in the room, there is less than a 1% chance of a shared birthday.

You can continue to calculate the chances of not having a shared birthday for any number of people:

1 x 365 / 366 x 364 / 366 x 363 / 366 x 362 / 366 . . .

Things change quickly as the number of people increases. With 10 people in the room, there is a better than 10% chance of a match. When there are 23 people in a room, the chance of a shared birthday is slightly greater than 50%, and it rises above 90% with 41 people.

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