19 replies to this topic

[bonanova]

There is a standard problem that asks, if you randomly break a stick into three pieces, the probability that the pieces can form a triangle. The answer is not unique because there are different ways to randomly create three pieces. The method that is the most challenging to analyze breaks the stick at a random point, randomly chooses one of the pieces, and breaks it at a random point. If you can solve that puzzle you may have insight to solve the following variant:

Break a stick at a random point. Break both pieces at random points. Randomly discard one of the pieces. What is the probability the remaining pieces can form a triangle?

[~andy~]

[Spoiler spoiler=please use spoilers]100%

[hhh3]

hmm.. well..

Spoiler for i would say

a probability of 1 bcoz i cant think of a scenario where 3 pieces of a stick cant mak a traingle

[phil1882]

if a side is longer than the sum of the other two it doesn't form a triangle.

Spoiler for my guess

after writing a cpu program, i get roughly 23%.

[hhh3]

oh.. yes.. thanks phil1882... i didnt really think about ""if a side is longer than the sum of the other two it doesn't form a triangle.
"" now it jst bcame a lot more complicated than it first seemed

[~andy~]

Yes, your'e right Phil, i didn't think of that either, without working it out it has to be less than 25%

[Prof. Templeton]

Spoiler for Some thoughts

the original problem, if I remember correctly, was more easily solved by evaluating points inside an equilateral triangle whose altitude represented the length of the stick. The three pieces would be the distance from the point to the three sides. A triangle can only be formed from the three pieces if two of them, when added, are not less then the third. The valid points inside the equilateral triangle where this occurs occupy 1/4 of the triangle (think Legend of Zelda Triforce). By discarding one of the pieces, I think we still have the same problem just with a smaller triangle. I will have to think about it more.

[CaptainEd]

Spoiler for My small simulation results

I don't feel up to analyzing this right now, so here's the result of 1764 cases. 0.219387755

[bonanova]

Hint:

Solve these problems first:

• Break the stick at two random points.
• Break the stick at a random point. Break one of the pieces at a random point.

These can be solved in closed form; are they the same?
Verify the answers by simulation if you like.

[bonanova]

Spoiler for My small simulation results

I don't feel up to analyzing this right now, so here's the result of 1764 cases. 0.219387755

That's close, Cap'n.
10x more cases should get you to 3-decimal-place accuracy.