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

1 Background

Our work was motivated by the following problem, which first appeared in a 1854 Examination atCambridge 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 1x y

0.24 0.43 0.33

The problem attracted the interest of the 19th century French probabilists Lemoine [5] andPoincare [8], was further studied in the mid 20th century by Levy [6], and was popularized byMartin Gardner [2]. It gives rise to the “broken stick model”, an important probabilistic model thatarises in areas ranging from biology (MacArthur [7]) to finance (Tashman and Frey [9]). The modelhas been shown to be a good match for a variety of real-world data sets, including intervals betweentwin births reported in the Champaign-Urbana News-Gazette, and intervals between aircraft crashesof U.S. Carriers (Ghent and Hanna [3]).

∗This project is part of an ongoing research project “Adventures with n-dimensional integrals”, carried out at theIllinois Geometry Lab, www.math.illinois.edu/igl. An expanded version of this report is expected to be preparedfor possible publication.

1

In our work we considered generalizations of the Broken Stick Problem to broken sticks with npieces.

2 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 pointschosen at random along its length. We are interested in forming triangles with the resulting npieces. 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 forma triangle?

In the case n = 3 there is one triple of pieces, and both questions reduce to the original BrokenStick Problem. Thus, the questions represent generalizations of this problem.

3 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 npieces that can form a triangle is

1 −n∏

k=2

k

Fk+2 − 1,

where Fn is the n-th Fibonacci number.

Theorem 2. Consider an n-piece broken stick. The probability that any triple of pieces chosenfrom the n pieces can form a triangle is

1(2n−2n

) .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 onetriple of pieces forms a triangle, and the second formula gives 1/15 as the probability that anytriple of pieces forms a triangle.

The values given by the theorem are consistent with those obtained from computer simulations.

4 Related Work and Future Directions

Lemoine [5] computed the probability for getting an acute triangle in the original 3-piece brokenstick problem.

Levy [6], among others, studied probabilistic aspects of the “broken stick distribution”, thedistribution of the lengths of the pieces in a randomly broken stick.

D’Andrea and Gomez [1] computed the probability of getting a polygon from an n-piece brokenstick.

2

A 2012 Putnam problem involved the following deterministic version of the Broken Stick Prob-lem with acute triangles: Given 12 real numbers the interval (1, 12), show that there exist three ofthese 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 anacute triangle with 3 of the pieces?

• Tetrahedra: What is the probability that, in an n-piece broken stick with n ≥ 6, one canform a tetrahedron with three of the pieces?

• Numbers of triangles: In an n-piece broken stick, the number of triangles that can beformed 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. Gomez, 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 ofrandom 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 probabilites, Bull. Soc. Math, de France 1 (1875), 39–40.

[6] P. Levy, Sur la division d’un segment par des points choisis au hasard, C. R. Acad. Sci. Paris208 (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. Poincare, Calcul des Probabilites, 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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