The names of 100 prisoners are placed in 100 wooden boxes, one name to a box, and the boxes are lined up on a table in a room. One by one, the prisoners are led into the room; each may look in at most 50 boxes, but must leave the room exactly as he found it and is permitted no further communication with the others.
The prisoners have a chance to plot their strategy in advance, and they are going to need it, because unless every single prisoner finds his own name all will subsequently be executed.
Find a strategy for them which has probability of success exceeding 30%.

Comment: If each prisoner examines a random set of 50 boxes, their probability of survival is an unenviable 1/2¹⁰⁰ ∼ 0.0000000000000000000000000000008. They could do worse—if they all look in the same 50 boxes, their chances drop to zero. 30% seems ridiculously out of reach—but yes, you heard the problem correctly!

It is not possible to have above 30% chance for all of them to find their own name. In fact, it is impossible to have above 30% chance for the first 2 people to find their own name. The best situation is that the first person picks 50 boxes and the second person picks the other 50 boxes. This has a probability of 25/99=0.252525 that they both find their own name.

Edited on April 11, 2017, 9:05 am

Posted by Math Man on 2017-04-11 08:20:11