I'm having trouble starting this problem.
Say you committed a crime and you're sentenced to jail. When entering the jail, you pick one ball from a box containing 3 balls, each numbered 0, 1, and 3. If you select 0, you get out of jail; if you select 1 or 3, you put the ball back into the box and after that number of years, you select again under the same conditions. You repeat this until you select a 0 and you're free to go. What is the expected value of how long you'll be in jail for?
The hint is that you're suppose to condition on the first ball selected. I don't see how this helps?