Length of broken stick, with a mark I can't seem to solve this problem. Any help is appreciated!
A person randomly put a mark on a specially designed stick of length L. The stick is designed
to be broken into two pieces randomly after 10 years of usage.
(a) What is the chance that the shorter broken piece contains the mark?
(b) What is the expected length of the broken piece that contains the mark?
(c) Given that the shorter piece contains the mark, what is the expected length of that piece?

a) The probability that the cut is on the shorter end is $P((X \leq Y \cap Y \leq 1/2)\cup (X \geq Y \cap Y \geq 1/2)) =  $
by symmetry
$$ 2P((X \leq Y \leq 1/2)) = 2 * 1/8 = 1/4 $$
(draw the unit square here to see this)
