The 1 meter length is unimportant.

I assume we're making two cuts uniformly distributed along the bar.

So let and be iid U(0,1) rvs.

I checked the three case and each case we want the largest cut to be less than .5.

The cuts are , and ,

BUT here I'm assuming which is really the order stats.

I'm trying to figure the logic here.

We need

The joint density of these two is