1. ## probability problem

A line segment of length 5 is broken at two random points along its length. What is the probability that the shortest of the three new segments has length longer than 1?
I'm going to assume that the distributions of the two points the segment is broken at are

a) independent

b) uniform[0,5]

Plotting this we see the following boolean contour plot, showing as a function of the break points $x,y$

whether the shortest segment has length less than 1.

those tannish triangles are the where the criterion fails to be met.

They have area $\dfrac{2\cdot 2}{2} = 2$ and there are two of them for a total area of $4$

The area of the probability space is $5\cdot 5=25$

so the probability that the criteria is met is

$P[\text{shortest segment has length <1}]=1 - \dfrac{4}{25} = \dfrac{21}{25}$

2. ## Re: probability problem

Thanks Romsek,
This is a very interesting question and that boolean contour plot looks like a great tool but I do not understand it.
Could you try and explain it to me.
I'd really appreciate that.
thanks

3. ## Re: probability problem

Originally Posted by Melody2
Thanks Romsek,
This is a very interesting question and that boolean contour plot looks like a great tool but I do not understand it.
Could you try and explain it to me.
I'd really appreciate that.
thanks
that's just a plot of the area on the joint distribution where the criteria is met and where it's not.

The tan triangles are where it's not.

The purple area divided by the total area is the probability of meeting the criteria of the length of the smallest segment.

We get away with such simple calculations because the joint distribution is uniform in both independent variables.

This turns out to be $\dfrac {21}{25}$

4. ## Re: probability problem

Got it! Thanks )