Section 9.2

Broken Sticks

Random cuts of a stick: forming a triangle, the shortest or longest piece, no piece exceeding a length.

Toolkit

Model a stick of length $1$ (or any fixed length) cut at uniform random points; break points are i.i.d. $\mathrm{Unif}(0,1)$ , giving a uniform point in the unit square (two cuts) or simplex (more pieces). Key facts:

Triangle inequality for pieces $a,b,c$ with $a+b+c=L$ : forms a triangle iff each piece is $<L/2$ (no piece exceeds the sum of the others).
Two cuts, uniform in $[0,1]^2$ : the “triangle” region has area $1/4$ .
$n$ independent uniform cuts split the stick into $n+1$ pieces with joint density uniform on the simplex $\sum x_i = 1$ ; by symmetry/exchangeability, $\mathbb{P}(\text{piece } i>t)$ depends only on $t$ via $(1-t)^{n}$ type volume ratios.
For $n$ exchangeable pieces each of mean $1/n$ : $\mathbb{P}(\min > d) = (1-nd)^{\,n-1}$ , valid for $0\le d\le 1/n$ .
Failure events for the triangle condition (one piece $>1/2$ ) are disjoint across pieces, enabling inclusion-exclusion via simple volume counts.

Problems