Two prisoners, two boxes, one look each

Question

Two prisoners, Ann and Ben. In a room are two closed boxes, one containing a slip reading "Ann" and the other a slip reading "Ben", placed in a random order. Each prisoner, alone and in turn, may open one box, then must close it and leave everything as it was. They cannot communicate after the first one enters. Both go free only if each finds the slip with their own name. 1. If each opens a box at random, what is the chance both succeed? 2. What is the best strategy, and what chance does it give?

Accepted Answer

Random guessing succeeds with probability 1/4. The best strategy succeeds with probability 1/2. Random guessing. Ann picks a box at random and finds her name with probability 1/2. Ben independently does the same. Both succeed with probability 1/2 × 1/2 = 1/4. The strategy. Label the boxes left and right. Agree beforehand: Ann opens the left box, Ben opens the right box. There are only two possible arrangements. If the slips are "Ann on the left, Ben on the right", both succeed. If they are swapped, both fail. So the chance of both succeeding is 1/2. Why that is a real improvement and where it…

Two prisoners, two boxes, one look each

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