Probability of their's constituting a triangle? 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 $p_1&lt;p_2$

notice $p_1&lt;.5$ and $p_2&gt;.5$ 

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 $|p_1-p_2|&lt;.5$.

So the answer is $\int_0^.5p_1 d p_1$ which is $\frac{1}{4}$
I'm sure there is a nicer solution:

label the points $0$ to $l$. And call the points $p_1$ and $p_2$ with $p_1&lt;p_2$

notice $p_1&lt;.5$ and $p_2&gt;.5$ 

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 $|p_1-p_2|&lt;.5$.

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