Thread: Some probability

A unit length is broken off into 3 pieces. What is the probability of them forming a triangle?



Moderator Edit: Approved Challenge question.

Originally Posted by dwsmith

A unit length is broken off into 3 pieces. What is the probability of them forming a triangle?



Moderator Edit: Approved Challenge question.

The length of the segment is irrelevant (provided it is nonzero).

Spoiler:

The distinction between and doesn't matter here.

WLOG let the segments be with .

Triangle inequality:

In fact this is equivalent to , for the given constraints. ( so we can substitute using .)

(Edited out a false start.)

So we have two variables x, y with uniform probability distribution in [0,1]. I made a sketch:



The shaded area is the probability we want, which is .

Originally Posted by undefined

The length of the segment is irrelevant (provided it is nonzero).

Spoiler:

The distinction between and doesn't matter here.

WLOG let the segments be with .

Triangle inequality:

In fact this is equivalent to , for the given constraints. ( so we can substitute using .)

(Edited out a false start.)

So we have two variables x, y with uniform probability distribution in [0,1]. I made a sketch:



The shaded area is the probability we want, which is .

Good job.

I let
,


The area of the region bounded is also one fourth of the triangle .