We have a line segments with length $l$ then you choose two random points and cut it from these points so that we have three piece of line segments.

What is the probability that these piece constitute a triangle ?

We need to use triangle inequality but I could not manage it. Thanks.


I'm sure there is a nicer solution:

label the points $0$ to $l$. And call the points $p_1$ and $p_2$ with $p_1<p_2$

notice $p_1<.5$ and $p_2>.5$

now, suppose $p_1$ has already been decided, then the fraction of points that would be suitable $p_2$'s is $p_1$ since we need $|p_1-p_2|<.5$.

So the answer is $\int_0^.5p_1 d p_1$ which is $\frac{1}{4}$

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