Recently, I devised the following 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.
Here’s how I solved this problem. Let be an indicator random variable for “no friend has a birthday on day , where stands for February 29 and stand for the “usual” 365 days of the year. Therefore, the quantity , representing the number of days of the year on which no friend has a birthday, can be written as
Let’s start with any of the “usual” days. In any four-year span, there are days, of which only one is February 29. Assuming the birthday’s are evenly distributed (which actually doesn’t happen in real life), the chance that someone’s birthday is not on day is
Therefore, the chance that all friends don’t have a birthday on day is
Since the expected value of an indicator random variable is the probability of the event, we see that
for . Similarly, the expected value for the indicator for February 29 is
Since even if and are dependent, we therefore conclude that
This function is represented by the red dots on the graph below.
In tomorrow’s post, I’ll calculate of the standard deviation of .