overlap, overstreched

An interesting challenge on The Riddler on the probability to see a random interval X’ing with all other random intervals when generating n intervals from Dirichlet D(1,1,1). As it happens the probability is always 2/3, whatever n>1, as shown by the R code below (where replicate cannot be replaced by rep!):

qro=function(n,T=1e3){
  quo=function(n){
     xyz=t(apply(matrix(runif(2*n),n),1,sort))
  sum(xyz[,1]<min(xyz[,2])&xyz[,2]>max(xyz[,1]))<0}
  mean(replicate(quo(n),T))}

and discussed more in details on X validated. As only a property on permutations and partitions. (The above picture is taken from this 2015 X validated post.)