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
In yesterday’s post, I began the calculation of the standard deviation of by first computing its variance. This calculation is complicated by the fact that are dependent. Yesterday, I showed that
To complete this calculation, I’ll now find the covariances. I’ll begin with if ; that is, if and are days other than February 29. I’ll use the usual computation formula for a covariance,
We have calculated earlier in this series. 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
so that the probability that no friend has a birthday on day is
Therefore, since the expected value of an indicator random variable is the probability that the event happens, we have
for . Therefore,
To find , we note that since is equal to either 0 or 1 and is equal to either 0 or 1, the product can only equal 0 and 1 as well. Therefore, is itself an indicator random variable, which I’ll call . Furthermore, if and only if and , which means that no friends has a birthday on either day or day . The chance that someone doesn’t have a birthday on day or day is
so that the probability that no friend has a birthday on day or is
Therefore, as before,
Since there are pairs so that , we have
The calculation of is similar to the above calculation; I’ll write this up in tomorrow’s post.