Unusual timing shows how random mass murder can be (or not)

This was one headline in the USA Today I picked from the hotel lobby on my way to Pittsburgh airport and then Toronto this morning. The unusual pattern was about observing four U.S. mass murders happening within four days, “for the first time in at least seven years”. The article did not explain why this was unusual. And reported one mass murder expert’s opinion instead of a statistician’s…

Now, there are about 30 mass murders in the U.S. each year (!), so the probability of finding at least four of those 30 events within 4 days of one another should be related to von Mises‘ birthday problem. For instance, Abramson and Moser derived in 1970 that the probability that at least two people (among n) have birthday within k days of one another (for an m days year) is

$p(n,k,m) = 1 - \dfrac{(m-nk-1)!}{m^{n-1}(m-nk-n)!}$

but I did not find an extension to the case of the four (to borrow from Conan Doyle!)… A quick approximation would be to turn the problem into a birthday problem with 364/4=91 days and count the probability that four share the same birthday

${30 \choose 4} \frac{90^{26}}{91^{29}}=0.0273$

which is surprisingly large. So I checked with a R code in the plane:

T=10^5
four=rep(0,T)
for (t in 1:T){
day=sample(1:365,30,rep=TRUE)
four[t]=(max(apply((abs(outer(day,day,"-"))<4),1,sum))>4)}
mean(four)


and found 0.0278, which means the above approximation is far from terrible! I think it may actually be “exact” in the sense that observing exactly four murders within four days of one another is given by this probability. The cases of five, six, &tc. murders are omitted but they are also highly negligible. And from this number, we can see that there is a 18% probability that the case of the four occurs within seven years. Not so unlikely, then.