This problem is known as the Monte Hall Problem. In this "Let's Make a Deal" scenario, a contestant picks one of three doors which might hide the "prize". The host Monte Hall knows the door that holds the prize, so after the contestant chooses a door he opens one of the other doors and shows that it does not have the prize. This leaves only two doors. He then asks the contestant if they wish to change their original choice and pick the remaining door.
This is the same problem as the prisoner example from the OP. So why didn't I just use the Monte Hall example above? Because I know you alcoholic drug using sorts. You want to
feel good now! You don't want to think too hard. If I would have called it the Monte Hall problem 80% of you would have Googled it and gotten the answer. I wanted you to THINK. OK enough ranting lol.
The best way (for me) to understand the best strategy (to stay or switch cells) is to do a decision tree. I'll use the Monte Hall example with Door #1 Door #2 and Door #3.
Plot the outcomes for the two different decisions (stay or switch) given that the prize "belongs to" door "A".
Lets call this real world #1 
This is where the prize is behind door number 2. You get the same win/loss ratio depending on whether you choose to stay with your first pick or switch. It's always better to switch because you will win 2 out of three times if you do.
There are also many youtube videos that explain this in a variety of ways. Just enter "Monte Hall problem". I found the above decision tree to be the best explanation. It's directly applicable to the prisoner example. It's twice as good for him to switch cells with B, as he will have a 2/3 chance of freedom as opposed to only 1/3 if he stays in cell A.
One more post to come.