Puzzler Answer, Encore 9, Prison Marbles
Ray: Many years ago there was this prisoner that was condemned to die.
Tom: It was actually my great-grandfather.
Ray: Right! And the warden came to visit him. And he said to him, "You're done for kid!"
Tom: "You are in big trouble."
Ray: "Tomorrow, you're history. However, being a betting man, I decided that I'm going to give you a fighting chance. You're odds of dying tomorrow are a 100 percent." Am I right or wrong?
Tom: You are right. "When the sun comes up tomorrow, man..."
Ray: "They are going to start warming up the electric chair and you're a goner. So I would like to increase your odds and make them 50 percent that you die and 50 percent that you live." And here's how he's going to do it is, he says: "I'm going to give you two shoe boxes. One box has 50 white marbles, and one box has 50 black marbles. And I'm going to come in tomorrow morning, just after they turn up the juice on the chair, and I'm going to pick a marble. I'm going to be blindfolded, so I can't see the marbles. And I'm going to pick a marble out. If it's white you live and if it's black, you're a goner."
Tom: "Now here's the thing. You can do anything you want with the 100 marbles. There are 50 black ones and 50 white ones. You don't have to leave them in the two shoe boxes as I have. You can put them all in one box. You can divvy them up in any way between the two boxes."
Ray: "But you must use all the marbles; you can't hide them."
Tom: You've got to use all one 100 marbles.
Ray: "Because when we are all done, and I have made my selection, we are going to count the marbles to make sure you've used all of them." So, of course, the person said, "Jeez, this is pretty good. I went from a 100 percent chance of dying down to a 50 percent chance of dying. Can I improve my chances any more?" Now, he was worried that he wouldn't get enough sleep that night, so he started working on it right away. The question is: Can he, in fact, improve his chances of surviving by any more than what they are?
Tom: And what is the best strategy for doing it? And there is no cheating allowed. All the marbles must be accounted for. You can't swallow them or paint them. So he's got to use all 100 of them and they'll be counted afterward.
Ray: In fact he can improve his chances by putting 99 marbles in one box, and one marble--a white one--in the other box. See right away when the warden comes in, he's got a 50-50 chance of picking a box. So if he picks the one with the white marble in it, then the fellow is automatically going to improve his chances to 75 percent. If he picks the other box, then his chances are still pretty close to 50-50. So by rearranging the marbles in this peculiar way he can improve his chances to almost 75 percent.
Tom: That's not bad. In fact his chances of losing, if I calculate correctly, are .05 times 50 over 99. That's 25 percent. A little more than 25 percent. That means that his chances of living are almost 75 percent.
[ Car Talk Puzzler ]
