the birthday problem [X’idated]

The birthday problem (i.e. looking at the distribution of the birthdates in a group of n persons, assuming [wrongly] a uniform distribution of the calendar dates of those birthdates) is always a source of puzzlement [for me]! For instance, here is a recent post on Cross Validated:

I have 360 friends on facebook, and, as expected, the distribution of their birthdays is not uniform at all. I have one day with that has 9 friends with the same birthday. So, given that some days are more likely for a birthday, I’m assuming the number of 23 is an upperbound.

The figure 9 sounded unlikely, so I ran the following computation:

extreme=rep(0,360)
for (t in 1:10^5){
  i=max(diff((1:360)[!duplicated(sort(sample(1:365,360,rep=TRUE))))]))
  extreme[i]=extreme[i]+1
  }
extreme=extreme/10^5
barplot(extreme,xlim=c(0,30),names=1:360)

whose output shown on the above graph. (Actually, I must confess I first forgot the sort in the code, which led me to then believe that 9 was one of the most likely values and post it on Cross Validated! The error was eventually picked by one administrator. I should know better than trust my own R code!) According to this simulation, observing 9 or more people having the same birthdate has an approximate probability of 0.00032… Indeed, fairly unlikely!

Incidentally, this question led me to uncover how to print the above on this webpage. And to learn from the X’idated moderator whuber the use of tabulate. Which avoids the above loop:

> system.time(test(10^5)) #my code above
user  system elapsed
26.230   0.028  26.411
> system.time(table(replicate(10^5, max(tabulate(sample(1:365,360,rep=TRUE))))))
user  system elapsed
5.708   0.044   5.762