The “birthday problem” is one of the classic problems in elementary probability because of its counter-intuitive solution. From Wikipedia:

In probability theory, the **birthday problem** or **birthday paradox** concerns the probability that, in a set of *n *randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year (except February 29) is equally probable for a birthday.

Recently, I devised the following different birthday problem:

Suppose that you have *n* friends, and you always say “Happy Birthday” to each friend on his/her birthday. On how many days of the year will you *not* say “Happy Birthday” to one of your friends?

Until somebody tells me otherwise, I’m calling this the *Facebook birthday problem* in honor of Facebook’s daily alerts to say “Happy Birthday” to friends.

In this series, I will solve this problem. While this may ruin the suspense, here’s a graph of the solution for along with error bars indicating two standard deviations.

Before deriving this solution, I’ll start with a thought bubble if you’d like to take some time to think about how to do this.

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