Another Prisonder Escape Puzzle (PENDING)

1. Problem Statement¹

You have got 100 prisoners who are allowed to strategies beforehand. One by one they will enter the room, pick 50 out of 100 boxes, and leave out the room. Once the prisoner has entered the room, they aren’t allowed to talk back (think of as keeping them in separate room afterwards). Inside the boxes are the numbers 1 to 100, placed randomly amongst the 100 boxes. Each prisoner is also labelled from 1 to 100. If each prisoner finds their label amongst the 50 boxes, they all will be freed. Navielly, the probability of this happening is \((1/2)^{100}\), which is practically 0. Can you improve this (by a lot!)?

2. Solution

Before I dwell into the answer, it is morally unfair for me to take the high ground. I heard the problem in a bit more forgiving terms. The same setup but basically with a hint

References

1. Veritasium: The Riddle That Seems Impossible Even If You Know The Answer ↩︎