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.
It is possible to have above 30% chance of success if each person picks 99 boxes. The first person picks every box except the first box, the second person picks every box except the second box, and so on. The probability that everybody finds their own name is approximately 1/e=0.367879.
Edited on April 11, 2017, 9:05 am
Posted by Math Man
on 2017-04-11 08:20:11