## Sunday, April 2, 2017

The birthday problem is a famous exercise in probability:  at a gathering of N people, what is the probability that two persons share the same birthday?  It seems counter-intuitive that when N = 23, the probability just exceeds 50%.  You can read the analysis in wikipedia, or my blog post about it (here).  Or how it snuck up on me in another context and I didn't recognize it (here).

Last week I was on Facebook (again), and thinking about birthdays.  I have a total of 45 friends who let Facebook know their birthday.  That's not many friends by FB standards, but I try not to take it personally.  In any event, I was curious to know how many shared birthdays there were among my FB friends.  It's easy to check once you're on the right page (thanks, Google, for pointing the way).  FB shows the birthday when you hover above the person's photo.  In any event, there were two pairs:  February 14 and April 11.

Here's the point:  it's easy to calculate the probability of one pair when N = 23.  It's much harder to enumerate the possibilities and their probabilities when N = 45 or 250 or whatever.  This is a job for simulation!

Fifty lines of Python (here) and I can run 10000 simulations for my case (N = 45) in an instant.  The results for one run show that the most common outcome is to have two pairs, and 80% of the runs have 1, 2, 3 or 4 pairs.

> python friends.py
2727 {2: 2}
2251 {2: 3}
1934 {2: 1}
1138 {2: 4}
591 {}
396 {2: 5}
248 {2: 1, 3: 1}
220 {2: 2, 3: 1}
175 {2: 3, 3: 1}
85 {3: 1}
83 {2: 6}
60 {2: 4, 3: 1}
17 {2: 5, 3: 1}
13 {2: 7}
9 {2: 2, 3: 2}
7 {4: 1}
6 {3: 2}
5 {2: 1, 4: 1}
3 {2: 6, 3: 1}
2 {2: 3, 4: 1}
1 {5: 1}
10000