Probability of their's constituting a triangle?

Question

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.

...or of http://math.stackexchange.com/questions/26424/ – Grigory M Jul 12 '14 at 20:28 — Grigory M, Jul 12 '14 at 20:28
...and see also http://mathoverflow.net/questions/2014/ – Grigory M Jul 12 '14 at 20:29 — Grigory M, Jul 12 '14 at 20:29
@GrigoryM: Thank you very much – mesel Jul 12 '14 at 20:31 — mesel, Jul 12 '14 at 20:31

Asinomás · Accepted Answer · 2014-07-12T20:37:09.250

-1

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 $\frac{1}{4}$

edited Jul 12 '14 at 20:37

answered Jul 12 '14 at 20:14

Asinomás

1 Answers1