I'm not sure the problem has a solution as stated. It contains internal inconsistencies.

--I will select one prisoner at random...
--I'm going to choose prisoners at random...
--I may choose the same guy three times in a row, or I may jump around and come back...
--given enough time, everyone will eventually visit the switch room as many times as everyone else...

If he is picking, and can elect not to choose prisoner X ever, it isn't random. And if the problem is inconsistent in its presentation, can it still have a logical answer?

I thought of it might involving odd/even parity, but if any one can visit the room multiple times that won't work...

Can our collective self-esteem wait a week for the answer?