n_sims <- 1e5

breaking_convention <- "all_at_once" # or stick_breaking

if (breaking_convention == "stick_breaking"){
  break_locations <- matrix(runif(n = n_sims), ncol=1)
  sorted_break_locations <- cbind(break_locations, matrix(runif(n = n_sims, min = break_locations, max=1), ncol=1))
} else {
  break_locations <- matrix(runif(n = 2*n_sims), ncol=2)
  sorted_break_locations <- t(apply(break_locations, 1, sort))
}

segment_lengths <- cbind(sorted_break_locations[,1],
                         sorted_break_locations[,2] - sorted_break_locations[,1],
                         1 - sorted_break_locations[,2])

sorted_segment_lengths <- t(apply(segment_lengths, 1, sort))

head(sorted_segment_lengths)

is_triangle <- function(x){
  as.logical(x[3] < (x[2] + x[1]))
}

triangles <- apply(sorted_segment_lengths, 1, is_triangle)

binom.test(x = rev(table(triangles)))