Date: Jun 5, 1996 5:39 AM
Author: Daniel A. Asimov
Subject: Re: probability of a triangle (SPOILER)
In article <9604298333.AA833373730@ccmail.odedodea.edu> Pat_Ballew@ccmail.odedodea.edu writes:

> a) If a unit length segment is randomly broken at two points along

> its length, what is the probability that the three pieces created in

> this fashion will form a triangle?

>

> Pat Ballew

> Misawa, Japan

> Pat_Ballew@ccmail.odedodea.edu

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Here's my answer to question a):

If we let T denote the planar triangle in 3-space whose vertices are

(1,0,0), (0,1,0), and (0,0,1), then choosing a point p at random on T

(i.e. with the probability of p lying in a subset S of T being proportional

to the area of S, or more precisely Prob(p is in S) = area(S) / area(T)

will simulate the random cutting of the unit segment desscribed above,

with resulting pieces of lengths x, y, and z respectively.

The condition that x, y, and z "will form a triangle" is equivalent to the

3 conditions: x + y > z, x + z > y, and y + z > x. The subset of T where

(x,y,z) satisfy these 3 conditions turns out to correspond to the little

triangle whose vertices are the midpoints of the edges of T. Therefore

Prob( the 3 pieces will form a triangle ) is one-fourth.

--Dan Asimov