Stick Broken Into Three Pieces

Source: http://www.cut-the-knot.org/

Problem:
Assume a stick is broken at random into three pieces. What is the probability that the pieces can form a triangle?

Solve it in following cases:
  1. Two break points are selected randomly (and distributed uniformly) on the stick.
  2. The stick is first broken into two pieces. The longest (or rather, not the shortest) is then broken into two.
  3. The stick is first broken into two pieces. A piece randomly selected with probability 1/2 is then broken into two.
  4. The stick is first broken into two pieces. A piece randomly selected with the probability proportional to its length is then broken into two. 
Update (Dec 22, 2010)
Solution posted by me in comments!

    Comments

    1. Pratik Poddar12:16 PM, December 22, 2010

      Solution at
      cut-the-knot website

      ReplyDelete
    2. TheorizerShiva1:23 PM, July 24, 2020

      1/4

      ReplyDelete
    3. Pratik Poddar1:15 PM, September 09, 2026

      1/4 is correct for the first scenario (two break points chosen independently and uniformly) - the breakable-into-triangle condition is that no piece exceeds half the stick, and the region in the unit square of break-point pairs where that holds has area 1/4. The other three scenarios give different answers, though - the second-breaking rule changes the distribution of piece lengths. Worth computing each! (replied using AI)

      ReplyDelete

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