What is the probability that a particular set of integer edge lengths selected from an interval $[1,n]$, can form a triangle? That is, let $a,b,c \in \{1,2, \dots n \}$. What is the probability that $a$, $b$ and $c$ are the side lengths to a triangle?
How might this extend to the case where one selects real number edge lengths from the unit interval? Can I look for a cube then exceed the pyramids from it?
Here is an except from my answer to a similar question:
If A+B > C, you have 100% chance of making a triangle. If A+B=C, the triangle is 2d and does not qualify. If A+B < C, you have three joined line segments that do not meet at two ends. You will have to figure the probabilities of each of these equalities/inequalities and go from there.
If A+B > C, you have 100% chance Not so, try $A=1, B=100, C=10$ for example.
– dxiv
Apr 30 '18 at 02:24
What about $n=2$? Note that all lengths will satisfy the triangle inequality except when we select exactly one length equal to $2$.
That is, $a=b=1, c=2$ fails the triangle inequality. It fails $3$ out of the $8$ cases.
– Mason Apr 30 '18 at 01:59