1. Probability as Problem SolvingThere are many charming probability problems that do not require any deep theory, although theymay require some cleverness. Here is a probability problem with a geometric flavor.Suppose a stick is randomly broken into three pieces. What is the probability that the three pieces canbe formed into a triangle?Solution RemarksThere is no loss of generality in assuming the stick has unit length. We take the term randomly to meanthat every triple of lengths (x, y, z) is as likely as any other triple of lengths. The set of all such triplescorrespond to the region of the plane x+y+z = 1 with x, y, z 2 0.The triples that are the sides of a triangle are a sub-region of this region; the required probability is justthe ratio of the area of this sub-region to the area of the plane region shaded in Figure 1.(0, 1,0)(1,0,0) , æ(0,0, 1)Figure 1: The Region r+y+z = 1.x, y, z > 0

Given that: A stick is randomly broken into three pieces. The objective is to find the probability…

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Suppose a stick is randomly broken into three pieces. What is the probability that the three pieces can be formed into a triangle?

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Counting And Probability

Section: Chapter Questions

Problem 7CC

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Question

1. Probability as Problem Solving
There are many charming probability problems that do not require any deep theory, although they
may require some cleverness. Here is a probability problem with a geometric flavor.
Suppose a stick is randomly broken into three pieces. What is the probability that the three pieces can
be formed into a triangle?
Solution Remarks
There is no loss of generality in assuming the stick has unit length. We take the term randomly to mean
that every triple of lengths (x, y, z) is as likely as any other triple of lengths. The set of all such triples
correspond to the region of the plane x+y+z = 1 with x, y, z 2 0.
The triples that are the sides of a triangle are a sub-region of this region; the required probability is just
the ratio of the area of this sub-region to the area of the plane region shaded in Figure 1.
(0, 1,0)
(1,0,0) , æ
(0,0, 1)
Figure 1: The Region r+y+z = 1.
x, y, z > 0

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