An ancient ruler often condemned to death the people he no longer favored. Howev
ID: 2959446 • Letter: A
Question
An ancient ruler often condemned to death the people he no longer favored. However, he amused himself by giving his victims a chance to live if they were both clever and lucky.The victim was given 6 vials filled with poison, 6 vials filled with water, and 2 boxes. He was to arrange the 12 vials in the 2 boxes. He could put any number he liked in each box, but he had to put at least one vial in each. After he had put all the vials into the boxes, the ruler would select one of the boxes and then draw out one vial from that box. The victim would have to drink its contents.
What arrangement of vials in the boxes gives the condemned man the best chance of avoiding the poison?
Explanation / Answer
The best outcome is when one box has 1 vial of water and the other box has the remaining vials. This is because 50% of the time the ruler will pick the box containing only the water and he will survive. The other 50% of the time the box with the 11 vials will be picked and he will have a 5/11 = 45.45% chance of survival, giving his overall chance of survival 50%*100% + 50%*45.45% = 72.725% chance of survival. Lets compare this case to the case where the boxes are distributed evenly with 3 poison/3 water per box. In that case, the ruler picks each box 50% of the time but each box carries with it a 50% chance of survival, meaning that his total survival odds are 50%*50% + 50%*50% = 25% + 25% = 50%. For the sake of trying it out, lets see what happens if we put one poison vial in one box and leave the other vials in the remaining box, the opposite of the choice we started with. In this case, 50% of the time the ruler will pick the box with the single poison vial, and our hero will die. The other 50% of the time the ruler will pick the box with the remaining vials and there will be a 6/11 or 54.54% chance of survival. Then the overall chance of survival is 50%*54.54% = 27.27% survival. So we have our answer!