The Broken Stick Problem

Dr Robert P. Batzinger, Payap University
10 March 2016

A classic problem in probability

Description of the Problem

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If a stick of length \( L \) is broken into three pieces \( (L1, L2, L3) \)
what is the probability that the three pieces will construct a triangle?

Properties of a Triangle

  • Sum of the size equals the length of the stick.
  • Each side is shorter than the sum of the other two sides.
  • The longest side must be less than half of the total length

\[ L = \sum_{i=1}^3 L_i \]

\[ L1 \lt L2 + L3\\ L2 \lt L1 + L3\\ L3 \lt L1 + L2\\ \]

\[ max(L1,L2,L3) \lt {L\over 2} \]

Modelling in R (L = 1)

ntries = 10000
cuts= cbind(runif(ntries,0,1),
  runif(ntries,0,1))

L1 = cuts[,1]            # 1st cut
L2 = (1 - L1) * cuts[,2] # 2nd cut
L3 = 1 - (L1 + L12)      # remainder

stickfrags = cbind(L1,L2,L3)

Graph of Chopped Sticks

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Sequences that Form Triangles

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Results of 5 runs

   Tri NonTri
1 1953   8047
2 1921   8079
3 1915   8085
4 1917   8083
5 2002   7998

Areas of Triangles Created

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Uniformly Distributed Cuts

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Normally Distributed Cuts

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