Prisoner Problem
Consider a prisoner who’s about to be put to death but is promised his freedom if he draws a silver ball from one of two identical vase. He’s allowed to put 50 silver and 50 gold balls among the two vase any way he likes. The vase are then going to be shuffled around out of his sight and he has to pick one vase and draw one ball at random from that vase.
So how does the prisoner maximize his chances of survival? If he puts an equal number of silver and gold balls into one of the urns, the other urn would also contain an equal number of silver and gold balls, and thus the probability of his drawing a silver would be one in two. Can these chances be improved? If so, how?
So how does the prisoner maximize his chances of survival? If he puts an equal number of silver and gold balls into one of the urns, the other urn would also contain an equal number of silver and gold balls, and thus the probability of his drawing a silver would be one in two. Can these chances be improved? If so, how?
Labels: logic
posted by ToM at
8:41 AM
14 Comments:
you put 25 silver in one vase then all the rest in the other vase so you then have them seperated so there is 25 silver in one side and 50 gold and 25 silver in the other. and i believe that that increses your chances to 2:3 i think.
25 gold in each on the bottom
25 silver in each on top.
no matter what he gets a vase with silver on top!
or maybe
put 1 silver balls in 1 vase
in other vase put 50 gold first, then the rest silver on top.
Put 1 silver ball in one vase, put the rest in the other. This increases chances to about 75%.
When the vases are "Shuffled around", does it shuffle the contents of the vases too?
I like phreebsd's soultion. 1 silver in one vase and 50 gold in the bottom of the other, followed by 49 silver on top. If the vases are not shaken too much, odds are, you'll draw a silver from the top.
If your allowed to examine the vases w/o seeing the contents, check which one weighs the least to find your 1 silver ball. Or cause a chip in the lip of one of the vases while putting the balls in, so you can tell them apart.
Otherwise, you still get slightly less that 75% chance.
i would think that nomatter what you do-the odds are 50/50
Knightmare:
If you split them up like earlier posted, 1 silver in one and the rest of the silver and all of the gold in the other, you would have more than 50/50. You have a 50/50 of choosing either the one silver or the 99 silver/gold. Then, if you choose the one without the single, you have a 49/99 to get it. That works out to be almost 50/50 for the second one.
So; with a 50% of choosing the single, which is 100% of getting silver, and a 49.49% of getting a silver out of the other one, you drastically improve your odds to nearly 75%(50% of the first one & 49.49% of 50% from the second one).
Chris; I would love to see what your opinion on this logic is, I'm not very good at calculating percents like this.
Hi last Anonymous. You flatter me. That could be a mistake when probability and statistics are involved.
Although there are lots of trick answers (definitely worth a shot in real life), I assume that all 100 balls must be used, and that the vases will be shaken thoroughly.
The 1 silver in one vase and everything in the other sounds good to me. Then the probability of picking silver is:
(1/2)*1 + (1/2)*(49/99) = 74/99 = 74.747474...%
Just split them half/half in each vase.
When the guards give you the shaken vases, you just pick one, spin it as fast as you can, and pick one in the middle.
Because of the centrifugal force, the gold balls, which are heavier will be pushed to the outside and in the middle there will only remain the silver balls.
(would also work with the 1/99 distribution mentioned earlier)
Ha ha .... Unique Answer... :)
Hi t:b:H. R U sure about that; it might behave more lie a whirlpool and put the gold balls in the middle ;)
I think that the best way is 50/50, put half god/silver in each vase, then there is a 50/50 chance of getting that silver ball
Put 25 gold on the bottom of each vase. Then put 25 silver balls on the top of the vase. That way you get a silver ball.
Corina, how can 50% be better that 74.7474...% ?
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