# It’s Follow-Up Friday: Birthday Puzzle edition!

Welcome to Follow-Up Friday!

By this time, you know the drill. Follow-Up Friday is a chance for us to revisit the subjects of previous posts and bring the PuzzleNation audience up to speed on all things puzzly.

And today, I’d like to return to the subject of birthday brain teasers!

Working on the Cheryl’s Birthday brain teaser a few days ago reminded me of another birthday-fueled puzzle that’s been around forever.

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.

Now, given that there are 365 days in the year, you’d assume the number of people necessary to get that probability of a shared birthday above 50% would be more than half of 365, or 183 people.

But it turns out that, statistically speaking, you don’t need anywhere near that many people.

Let’s break it down. Person A has a birthday. Person B has a birthday. There’s only one possible pairing, A-B. Person C has a birthday, but creates three possible birthday pairings: A-B, A-C, and B-C.

Person D could have a different birthday, but the introduction of Person D begins escalating the number of POSSIBLE shared birthdays. With these four people, we have SIX possible pairings: A-B, A-C, A-D, B-C, B-D, and C-D.

Our fifth person, Person E, allows for TEN possible pairings: A-B, A-C, A-D, A-E, B-C, B-D, B-E, C-D, C-E, and D-E. The probability of a shared birthday is increasing much faster with each new person.

As it turns out, it only takes 23 people to give us a 51% probability of a shared birthday.

And that would certainly save on catering.

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