An n-gon from a broken stick, Geometric Probability

Problem 9.1

For an integer $n \geq 3$ , a unit stick is broken at $n - 1$ independent uniform random points, producing $n$ pieces. What is the probability that the $n$ pieces can be reassembled to form a (non-degenerate) $n$ -gon?

Solution. Let $s_1, s_2, \ldots, s_n$ be the piece-lengths, so $\sum s_i = 1$ and $s_i \geq 0$ . The pieces can form a non-degenerate $n$ -gon iff each is less than the sum of the others: $s_i < \sum_{j \neq i} s_j = 1 - s_i \;\;\iff\;\; s_i < \tfrac{1}{2} \text{ for every } i.$

We compute $\mathbb{P}(\text{some } s_i \geq \tfrac12)$ by inclusion–exclusion. Let $A_i = \{ s_i \geq \tfrac12 \}$ . Each piece $s_i$ marginally has density $(n-1)(1-s)^{n-2}$ on $[0, 1]$ (Beta $(1, n-1)$ ), so $\mathbb{P}(A_i) = \int_{1/2}^{1} (n-1)(1-s)^{n-2} \, ds = (1 - \tfrac12)^{n-1} = \frac{1}{2^{n-1}}.$

The events $A_i$ are mutually exclusive (two pieces cannot both exceed $\tfrac12$ , since their sum would exceed $1$ ). Hence by addition, $\mathbb{P}(\text{some } s_i \geq \tfrac12) = n \cdot \mathbb{P}(A_1) = \frac{n}{2^{n-1}}.$

Therefore $\boxed{\mathbb{P}(n\text{-gon}) = 1 - \frac{n}{2^{n-1}}.} \qedhere$

For $n = 3$ (triangle) the probability is $\tfrac14$ ; for $n = 4$ (quadrilateral) it is $\tfrac12$ ; for $n = 5$ (pentagon) it is $\tfrac{11}{16}$ . The probability tends to $1$ as $n \to \infty$ — intuitively, when a stick is broken into many pieces, “short enough pieces” becomes automatic. This generalises the $n = 3$ case of Chapter 2 (whose complement is the $n = 3$ instance here).

The result is cited frequently on math.stackexchange, usually as a classroom exercise illustrating inclusion–exclusion in the simplex.

import numpy as np
# Break stick at n-1 uniform points; check if all n pieces < 1/2.
for n in [3, 4, 5, 6]:
    pts = np.sort(np.random.rand(n-1, 10**6), axis=0)
    spacings = np.vstack([pts[:1], np.diff(pts, axis=0), 1 - pts[-1:]])
    ngon = (spacings < 0.5).all(axis=0)
    print(f"n={n}: sim={ngon.mean():.5f}   exact={1 - n/2**(n-1):.5f}")
# n=3: sim=0.24987  exact=0.25000
# n=4: sim=0.49910  exact=0.50000
# n=5: sim=0.68732  exact=0.68750
# n=6: sim=0.81286  exact=0.81250