Unusual timing shows how random mass murder can be (or even less)

This post follows the original one on the headline of the USA Today I read during my flight to Toronto last month. I remind you that the unusual pattern was about observing four U.S. mass murders happening within four days, “for the first time in at least seven years”. Which means that the difference between the four dates is at most 3, not 4!

I asked my friend Anirban Das Gupta from Purdue University are the exact value of this probability and the first thing he pointed out was that I used a different meaning of “within 4”. He then went into an elaborate calculation to find an upper bound on this probability, upper bound that was way above my Monte Carlo approximation and my rough calculation of last post. I rechecked my R code and found it was not achieving the right approximation since one date was within 3 days of three other days, at least… I thus rewrote the following R code

T=10^6
four=rep(0,T)
for (t in 1:T){
  day=sort(sample(1:365,30,rep=TRUE)) #30 random days
  day=c(day,day[day>363]-365) #account for toric difference
  tem=outer(day,day,"-")
  four[t]=(max(apply(((tem>-1)&(tem<4)),1,sum)>3))
  }
mean(four)

[checked it was ok for two dates within 1 day, resulting in the birthday problem probability] and found 0.070214, which is much larger than the earlier value and shows it takes an average 14 years for the “unlikely” event to happen! And the chances that it happens within seven years is 40%.

Another coincidence relates to this evaluation, namely the fact that two elderly couples in France committed couple suicide within three days, last week. I however could not find the figures for the number of couple suicides per year. Maybe because it is extremely rare. Or undetected…

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

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

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.

Terrible graph of the weekend

Every week, the weekend edition of Le Monde contains a tribune around some statistics and almost irremediably the illustrations are terrible. Witness the one above where the move from 940 millions to more than a billion is completely out of proportions, both for the disk areas and the number of people! Same thing for the split pie chart below:

What’s wrong with a regular pie-chart?! And the increase in disabled planes below is not much better, moving from 5 to 8 plane silhouettes… The illustrators should read Tufte’s books instead of USA Today…!

Ps-I seem to be always picking at Le Monde, but this is only because I have a subscribtion to its weekend edition!