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The Birthday Problem

With 365 potential birthdays per year, how many people do you have to put in a room at random until there are two or more with the same birthday?

Most people’s quick intuition will lead them to think that with 365 days in a year there would need to be somewhere around that many people or more to find two matching birthdays.

However, with just 23 people in the room there is a greater than 50% chance that two or more people’s birthdays match.

The birthday problem or birthday paradox is an interesting look at statistics that seemingly defy logic. It’s not really a paradox to people with a firm grasp of statistics such as the Wolfram Alpha explanation of this problem, but to most it’s a little confusing at first.

birthday problem

With just 23 people chosen at random there is about a 50.73% probability that two of them share the same birthday, and with only 57 people there is a 99% chance of a birthday match. This assumes birthdays are evenly distributed throughout the year and ignores leap day.

With a probability problem like this you must consider all the potential combinations of birthdays, not just the total number of birthdays. There are 23 people with birthdays, and each one of them may be a match to 22 other people; the math says there are 253 different possible pairs of birthdays. Finding at least two that match doesn’t seem so unlikely now.

The almost birthday problem shows the probability that two people will have a birthday within one day of each other. For a match with this problem, only 14 people are needed to reach a probability greater than 50%.

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