A stick is broken at two points, at random. If the length of the stick is 6, then what is the probability that all three resulting pieces are shorter than 3 units?

I'd use contour probability mapping

Let one piece be x another y-x and the third 6-y

we know

0

y-x>0 which simplifies to y>x

y-x<6 which simplifies to y

and

0<6-y<6 which simplifies to -6<-y< 0 or better still 0

Plot all those lines on a number plane.

The area in the middle conatins all the possible outcomes.

Now you want the section inside where

x<3

y-x<3 and

6-y<3

Find the area contained in this smaller region

Put the little area over the big area and you have your probability.