constantin

A stick (let’s say a safely match) is broken in four points so five segments are generated, the question is what is the probability that with these segments to be able to construct a pentagon. I solved the problem for two points (triangle) and for three point (quadrilateral) using 2D and 3D representation but I think that for more dimensions it isn’t a good approaching for this problem.

Denis

Too headachy for me!
But for something that would be reasonably close:
_._._._._._._._._._
10 unit stick, with 9 spots (the dots) from which to pick 4 at random.
Resulting lengths of the 5 pieces:
1 1 1 1 6 : no
1 1 1 2 5 : no
1 1 1 3 4 : yes
1 1 2 2 4 : yes
1 1 2 3 3 : yes
1 2 2 2 3 : yes
2 2 2 2 2 : yes
Probability = 5/7

Hoempa

Isn't this equivalent to asking "what is the probability that the longest segment is larger than half of the length of the stick?"?

Denis

Agree, but larger or equal

Hoempa

Hm yes but it wouldn't matter for the outcome I believe. Maybe that's too much of a side question.

constantin

Something like this but "less than half of the stick"

Denis

Agree; that's what I sais in my last reply: "Agree, but larger or equal".