What is behind door #3?

Question

Here is a trivia quesion I heard recently:

There are three doors.  One door has a prize, the other 2 doors have lemons.  You choose a door (but don't open it)  and there are two doors left.  A friend comes along and opens one of the remaining doors revealing a lemon.   What is the probability that the remaining unchosen door has the prize?  What is the probability that the door you first chose has the prize?  Do you stick with the door you first picked or do you pick the remaining door?  Demonstrate the answer mathematically.