Download Random Points, Broken Sticks, and Triangles Project Report

Random Points, Broken Sticks, and Triangles
Project Report∗
Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Aananya Uppal,
A.J. Hildebrand (Faculty Mentor)
University of Illinois at Urbana-Champaign
April 25, 2013
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Background
Our work was motivated by the following problem, which first appeared in a 1854 Examination at
Cambridge University.
Broken Stick Problem. A stick is broken up at two points, chosen at random along its length.
What is the probability that the pieces obtained form a triangle?
0
x
0.24
y
0.43
1
0.33
The problem attracted the interest of the 19th century French probabilists Lemoine [5] and
Poincaré [8], was further studied in the mid 20th century by Lévy [6], and was popularized by
Martin Gardner [2]. It gives rise to the “broken stick model”, an important probabilistic model that
arises in areas ranging from biology (MacArthur [7]) to finance (Tashman and Frey [9]). The model
has been shown to be a good match for a variety of real-world data sets, including intervals between
twin births reported in the Champaign-Urbana News-Gazette, and intervals between aircraft crashes
of U.S. Carriers (Ghent and Hanna [3]).
∗
This project is part of an ongoing research project “Adventures with n-dimensional integrals”, carried out at the
Illinois Geometry Lab, www.math.illinois.edu/igl. An expanded version of this report is expected to be prepared
for possible publication.
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In our work we considered generalizations of the Broken Stick Problem to broken sticks with n
pieces.
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The Broken Stick Problem with n Pieces
To obtain an n-piece version of this problem, we consider a stick that is broken up at n − 1 points
chosen at random along its length. We are interested in forming triangles with the resulting n
pieces. There are two natural questions one can ask.
Question 1. What is the probability that there exist three of the n pieces that can form a triangle?
Question 2. What is the probability that any triple of pieces chosen from the n pieces can form
a triangle?
In the case n = 3 there is one triple of pieces, and both questions reduce to the original Broken
Stick Problem. Thus, the questions represent generalizations of this problem.
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Results
We were able to answer Questions 1 and 2 exactly by the following theorems.
Theorem 1. Consider an n-piece broken stick. The probability that there exist three of the n
pieces that can form a triangle is
n
Y
k
1−
,
Fk+2 − 1
k=2
where Fn is the n-th Fibonacci number.
Theorem 2. Consider an n-piece broken stick. The probability that any triple of pieces chosen
from the n pieces can form a triangle is
1
2n−2 .
n
For n = 3 both formulas reduce to 1/4, which is consistent with the solution to the original
Broken Stick Problem. For n = 4, the first formula gives 4/7 as the probability that at least one
triple of pieces forms a triangle, and the second formula gives 1/15 as the probability that any
triple of pieces forms a triangle.
The values given by the theorem are consistent with those obtained from computer simulations.
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Related Work and Future Directions
Lemoine [5] computed the probability for getting an acute triangle in the original 3-piece broken
stick problem.
Lévy [6], among others, studied probabilistic aspects of the “broken stick distribution”, the
distribution of the lengths of the pieces in a randomly broken stick.
D’Andrea and Gómez [1] computed the probability of getting a polygon from an n-piece broken
stick.
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A 2012 Putnam problem involved the following deterministic version of the Broken Stick Problem with acute triangles: Given 12 real numbers the interval (1, 12), show that there exist three of
these numbers that form the lengths of an acute triangle.
Questions we plan to investigate in future work include:
• Acute triangles: What is the probability that, in an n-piece broken stick, one can form an
acute triangle with 3 of the pieces?
• Tetrahedra: What is the probability that, in an n-piece broken stick with n ≥ 6, one can
form a tetrahedron with three of the pieces?
• Numbers of triangles: In an n-piece broken stick, the number of triangles that can be
formed ranges from a minimum of 0 to a maximum of n3 . What is the probability distribution
of the number of triangles that can be formed?
References
[1] C. D’Andrea and E. Gómez, The broken spaghetti noodle, Amer. Math. Monthly 113 (2006),
555-557.
[2] M. Gardner, The colossal book of mathematics, W. W. Norton, New York, 2001. (Chapter 21)
[3] A. W. Ghent and B. P. Hanna, Application of the “Broken Stick” formula to the prediction of
random time intervals, American Midland Naturalist 79(2) (April 1968), 273–288.
[4] G. Goodman, The problem of the broken stick reconsidered, Math. Intelligencer 30 (2008),
43–49.
[5] I. Lemoine, Sur une question de probabilités, Bull. Soc. Math, de France 1 (1875), 39–40.
[6] P. Lévy, Sur la division d’un segment par des points choisis au hasard, C. R. Acad. Sci. Paris
208 (1939), 147–149.
[7] R. H., MacArthur, On the relative abundance of bird species, Proc. Nat. Acad. Sci. USA 43
(1957), 293–295.
[8] H. Poincaré, Calcul des Probabilités, Gauthier-Villars, 1912.
[9] A. Tashman, R. J. Frey, Modeling risk in arbitrage strategies using finite mixtures, Quant.
Finance 9 (2009), 495–503.
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