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Topic: Easy: Marble Jars (Read 8453 times) |
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ootte
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Easy: Marble Jars
« on: Jul 24th, 2002, 12:54am » |
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Well. Although the answer looks obvious, I'm sure it is wrong.
The aim of redistributing marbles seems to have roughly the same amount of white marbles in each jar, because this means surviving.
Without redistributing the marbles you have a 50% chance of surviving (= you are presented the jar with only white marbles in it).
With redistributing you can make each jar have 25 white marbles in it. So, it doesn't matter with jar is presented. But the chance of surviving is still 50% (choosing one of 25 white marbles out of 50 marbles). So, that doesn't help much.
If you put only one white marble into one jar, and all others marbles in the second jar, the chance of surviving is: 100% when you are presented the first jar, and 49.49% when the second one is presented.
That's the best you can get, in my opinion.
Comments are appreciated.
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Oliver
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James G
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Oh, I see. I went on to the next question after I decided to put 25 black marbles at the bottom of each jar, and 25 white ones at the top... but I guess that's not allowed.
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ootte
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Re: Easy: Marble Jars
« Reply #2 on: Jul 24th, 2002, 1:10am » |
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on Jul 24th, 2002, 1:04am, James G wrote:| after I decided to put 25 black marbles at the bottom of each jar, and 25 white ones at the top... but I guess that's not allowed. |
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That doesn't violate the rules, and you get a much better chance of surviving. You only need to make sure that the black marbles are fully covered with with marbles.
I wonder what the 'right' solution (that would be given by the author) is ...
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Oliver
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Gabin Kattukaran
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i think that you should move all but one white marble to the jar with black ones. that way, if you get the (original) white jar, you're sure to live and if you get the other one, you've got a 50% chance of living.
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Josh
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on Jul 24th, 2002, 1:04am, James G wrote:| Oh, I see. I went on to the next question after I decided to put 25 black marbles at the bottom of each jar, and 25 white ones at the top... but I guess that's not allowed. |
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That's what I'd do. With the black marbles at the bottom of the jars, the chance of survival would be almost 100% as it would be nearly impossible to accidently pick a marble out of the bottom half of the jar.
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Alex
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Even better: Just keep redistributing marbles for all eternity. It specifically states there is no time limit. 
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Ion Rush
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yes, without the possibility that the guards will give a really hard shake, layering white on top of black would give 100%
how would the guards react if you put all 100 marbles in one jar, and then they presented you with the empty jar to draw a marble from. would the guards get executed? would drawing nothing not count as living by default, (you didn't get black) which means if you did get the mixed jar, being that it has 100, you move from 49.49% to 50%
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James G
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True, the guards could shake Re: Easy: Marble Jars
« Reply #7 on: Jul 31st, 2002, 12:43am » |
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True, the guards could shake the jars. But, imagine this scenario...
The prisoner has just put one white marble in one jar, and all the other marbles in the other. They are blind-folded, and reach into the jar they are given, finding to their releif that it contains only one marble, which they know must be white. Then they remove their blind fold, and discover they are holding a black marble, and the other jar also contains just one black marble. The guards have a good laugh, and one of them says "That joke always cracks me up."
The problem does state that the jar will be selected randomly, but it doesn't specifically state that the guards won't change the contents after the prisoner has been blind-folded. 
Ok, that probably doesn't actually add anything to the discussion, but I just thought it was amusing in a sadistic kind of way.
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Brion
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Re: Easy: Marble Jars
« Reply #8 on: Jul 31st, 2002, 3:02pm » |
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The layering was my original answer, but Mr. Wu has clarified (in a separate topic) that the jars would be shaken after the sorting has taken place. I would also assume the jars are not small mason jars, but something larger so assuming that packing a single jar with all 100 marbles, white on top, would disallow room for jumbling when shaken can't work either.
I believe 1 white marble in one jar and 99 mixed marbles in the second jar provides you with a 75% chance of living. How?
If you pick the jar with only 1 white marble, your chance is 100%. If you pick the jar with 99 mixed marbles, your chance is slightly less than 50%. The combined chance of survival is slightly less than 75%.
Brion
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Donkeyboy
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I am definately no expert but....
I would have said DO NOT redistribute the marbles.
If you leave all white marbles in one jar and all black marbles in the other, you have a 50% chance of surviving.
If you redistribute the marbles, you will be effectively *reducing* your chance of choosing correctly, since if you are lucky enough to get the jar with the most white marbles, you *still* have to pick a white marble from the jar.
For example, if you swap one white marble and one black marble you have either a 2% chance of living or a 98% chance of living, depending on the jar you get. BUT, you still have to take into account the 50% chance of getting either jar to start with. So, by my very freaky, crack-induced logic, if you get the 49 white marble jar, you have a 98%-of-50% chance of living (which to me eqauls a 49% chance overall).
if you were to swap 25 white marbles with 25 black marbles, then both jars are identical, and you *Still* have a 50% chance of survival.
If you don't shuffle any of the marbles, you stay at 50%.
In fact , the more I think about it, the more it appears to me that the contents of the jars are irrelevant. You start at a 50% chance of survival and only go downhill from there if you choose to redistribute the marbles.
Any one else follow what I'm trying to say ?
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william wu
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Re: Easy: Marble Jars
« Reply #10 on: Jul 31st, 2002, 11:12pm » |
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You argument sounds fine, but you are not considering the option of simply moving one marble to another jar without exchange. The problem says that you can do whatever you want with the marbles, as long as when you're done, every marble is in one of the two jars. So you can put a white marble in the predominantly black jar, and NOT put a black marble in the predominantly white jar. They don't all have to be swap operations. As you have observed, swaps are useless.
The key thing to notice with this problem is that it's redundant to have 50 white marbles in a purely white jar. Suppose you get that purely white jar. Does it make any difference whether there are 1,000,000 white marbles inside, or just 1? No, it doesn't. After you received that jar, you will pull out a white marble with 100% certainty. So you might as well just leave only one white marble in one jar, and put all your other white marbles in the predominantly black jar, so you can improve your chances of survival in the unfortunate event that you get the nonwhite jar. This stategy gets you a nearly 75% chance of survival.
Hope that helps!
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| « Last Edit: Jul 31st, 2002, 11:13pm by william wu » |
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Donkeyboy
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aaaaahhhhhh
You're right.....I was assuming that marbles needed to be swapped.
Thanks for clearing that up
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James Paige
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Igor sat bound hand and foot on the hard wooden chair in the darkened room. The taller kidnapper, the one they called Marco, placed two large jars on the big hardwood table in the middle of the room.
The other kidnapper, the one called Lupo sniffed a bit, took a last drag from his failing cigarette, and tossed the butt at Igor. It bounced lightly off his chest, releasing a little puff of harmless sparks.
"This is a little game we like to play sometimes," said Lupo. Lupo was the only one who ever addressed Igor directly. "This is for that we have a little fun, and that we show you we are not such bad guys, and give you a chance to live."
Igor listened intently. The bruises on his body throbbed, where they had beaten him, and though he believed none of it, this talk of a chance to live caught his full attention.
"We have here two jars of marbles." explained Lupo. "One with fifty white marbles, one with fifty black marbles."
Igor looked at the jars on the table. They were directly under the single bare lightbulb that lit the room. The bottom of each was full of shinly lumps, white in the jar on the left, black in the jar on the right.
"We are going to put the blindfold back on you, and you will pick one marble from a jar we will give to you at random. If you pick a white marble, you live. And if you pick a black marble," Lupo smiled darkly, "you die."
Marco chuckled a little, standing with his huge arms crossed over his chest.
Igor's heart sank. This was like flipping a coin for his life.
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James Paige
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"But there is more!" said Lupo brightly, like he was pitching a kitchen-gadget on the infomercial channel. "Before we blindfold you, we let you mix up the marbles any way you want, just so long as they all end up in the jars."
"I can redistribute them any way I like?" asked Igor hollowly.
"Yeah," said Lupo. "But we shake them up before we let you pick a marble, so it won't do no good to put all the white ones on top. You gotta play fair."
Marco chuckled again.
Igor did not like the sound of this. It did not make sense. Why would men who had interrogated him for hours, beaten him black and blue, and left him with nothing to eat and just an occasional mouthful of water for days suddenly give him a chance to go free? And why would they do it with such a silly little game?
"Move him over." said Lupo.
Marco unfolded his arms, and dragged Igor's chair over to the table and positioned him in front of the jars.
"Untie his hands." said Lupo.
As Marco complied, Igor felt the blood surge back into his hands. How long had he been tied? he wondered. He immediatedly began massaging his wrists. They had been numb before, and now they hurt so suddenly, but he did not care. It felt wonderful to have them free.
"Take your time." said Lupo. "Think about it carefully."
Igor stared at the jars, still massaging his hands. This was just simple probability. It wasn't so hard. He tought about it for a moment, and finally he reached into the left jar and took out a handfull of white marbles. He looked at them in his hand. Eight marbles. He dropped all of them into the black jar.
Lupo and Marco had been standing silently, watching him, but now Lupo leaned over to Marco and whispered something. Marco nodded and left the room.
Igor took another handful of the white marbles, and dropped them into the black marble jar. He knew if he was given a jar with only white marbles in it, he was sure to live, and if he was given a jar with a mixture of black and white marbles, at least he still had a chance. He would leave as few marbles as possible in the white jar. In fact, he would just leave one. Lupo had not said that was against the rules. Then he would put all the rest of the marbles in the other jar. One jar of guaranteed life, and one jar of nearly-fifty-fifty chances. Taken together that would give him almost seventy-five percent chances of living... assuming that the kidnappers would be honest about randomly picking a jar for him to draw from.
Igor's hand stopped as he hovered his fourth handful of white marbles over the black marble jar. How were they going to choose randomly? Would one of them be blindfolded and shuffle the two jars around and then pick one? Certainly not! They would just pick one and call it random. They would never give him the jar with one white marble in it. This was a trick. This was a joke. How could he trust men like this to play fair? He withdrew his hand from the jar, and stared at the marbles.
Lupo snorted, and turned to light another cigarette.
They would not give him the jar with one white marble, they would give him the jar with fifty black marbles and forty-nine white marbles. They would shake the jar up, and he would draw, and his chances of living would be just slightly less than fifty-fifty. What else could he do? After a moment of thought, he dug deep into the black marble jar, and pulled up a handfull of mostly black marbles and dropped them into the white jar. If he put twenty-five of each color in each jar then his chances would truly be fifty percent. He would take the randomness out of their hands completely. If he was to face half a grim reaper he would face it with his own luck.
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James Paige
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Igor tried to remember exactly how many white marbles he had moved to the black marble jar. The first handful had been eight. The second had been seven. Fifteen. He had not counted the third handful, had he? Now he would have to count them all to be sure he had exactly the right number in each to insure fifty-fifty-- Igor stopped again. Fifty fifty wasn't enough. This was a toss of a coin. Why do with marbles what could be done with a coin? A chill ran down his spine. He had let himself be carried away in the logic of the game, and that was a mistake. He was forgetting who he was dealing with. Cruel evil men. This was better than a coin toss for them because it gave him a glimmer of hope. It let him think he had a chance to save himself, or at least to improve his odds. Surely this was all a lie. The were going to kill him one way or the other, and this was just them having one last laugh at his expense. Both his hands were full with marbles now, and he dropped them all into the right-hand jar. He would not do it. He would just sit here and they-- Wait.
Igor thought about his hands. His hands were free. It was the same ropes which had bound his hands that bound him to the chair. His ankles were still tied, but he was not tied to the chair anymore. His hands were completely free, and there was nobody in the room except Lupo.
He glanced up at Lupo, careful not to move his head. Lupo was now pacing patiently up and down the room on the other side of the big table, flicking his cigarette up and down between his teeth. He could see the strap of Lupo's gun holster over his shoulder, but Lupo's hands were clenched behind his back. Igor knew he had to take a chance. This wasn't the kind of chance you could use logic or probability on, this was just the kind of chance you had to take when there was no other way.
Igor gently reached for the right-hand jar and withdrew a single black marble. He rolled it between his fingers and watched Lupo pace, down, and then up, and then down.
"I have a question." said Igor.
Lupo started, and looked at Igor with raised eyebrows.
Igor threw the marble with all his might. It struck Lupo directly in the eye, and he reeled and stumbled back, uttering a stream of curses punctuated by death threats.
Igor siezed the black marble jar and smashed it on the edge of the table. Glass and marbles flew everywhere, and he felt pain in his fingers, but he ignored it. Picking up the largest shard of the jar he toppled sideways off the chair and immediately began slicing franticly at the cords that bound his ankles.
Black and white marbles spread from the tabletop, rolling of the edges and bouncing on the floor, clinking like metal on the bare concrete. Lupo, one hand over his eye, ceased his stream of profanity to call loudly for Marco, and staggered forward, drawing his gun.
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James Paige
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The cord frayed and snapped, and Igor's feet were free.
Truly the white marbles were life. Lupo's foot trod on a pair of them, and slipped from underneath him. He fell hard, the gun discharging into the air. The sharp noise of the gunshot made the next few seconds seem deathly silent, while Lupo strugged to get up, and Igor scampered forwards on hands and knees, and then the brief silence was shattered by Lupo's scream as Igor drove the glass shard deep into his outstretched leg.
Both of Lupo's hands clutched the wound. Where was the gun?
Igor jumped to his feet, unsteadyly, as his ankles experinced the same burning that his wrists had just minutes ago. Where was the gun? No time to look for the gun. Marco would be coming.
Igor still had the big shard of the black marble jar in his hand. No sooner had he positioned himself by the door, than Marco entered in a dead run. Without time to think, Igor lunged, putting his full weight behind the shard.
Igor choked on his breath as he watched Marco slump to the ground. he had nearly taken his head off. He felt the warm sticky wet blood splattered all the way up his arm. Marco's blood. Lupo's blood. Some of it was his own blood. He could now feel the sharp sting in his hands where he gripped the shard. He had never killed anyone before. He had never even really imagined it.
Lupo was cursing loudly again. The gun was still somewhere on the floor. Igor ran out the door. Where were the two other guys? The ones who had been there when he was interrogated? Were they even still around? Igor realized he had dropped the shard of glass. He wished he still had it. No time to go back for the shard. No time to go back for the gun. Can't go back in there. There is death in there. There is death in there.
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Jeremiah Smith
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Re: Easy: Marble Jars
« Reply #16 on: Aug 9th, 2002, 12:47am » |
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Good story. 
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Mongolian_Beef
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Re: Easy: Marble Jars
« Reply #17 on: Aug 13th, 2002, 8:24pm » |
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it seems the answer has been given above by the administrator but this is just a thought on why the answer would need to turn out that way.
theres been a lot of talk about having 25 blacks and 25 whites in either jar. if you did nothing youd have that same 50% probability. you could also get the same 50% probability by moving 49 whites and 1 black. so one jar would have 2 marbles the other 98.
the point is, however, youre guaranteed at least 50% odds if one jar only has white marbles. then any white marbles moved to the other jar, helps maximize your odds. you can throw 49 white marbles into the black jar while still maintaining one pure white jar. that gives you the best odds.
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Gleb
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on Jul 25th, 2002, 3:32pm, Josh wrote:
That's what I'd do. With the black marbles at the bottom of the jars, the chance of survival would be almost 100% as it would be nearly impossible to accidently pick a marble out of the bottom half of the jar. |
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avjewe
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Re: Easy: Marble Jars
« Reply #19 on: Oct 2nd, 2002, 6:22pm » |
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Depending on the weather, there's an extra trick.
1) Lay all of the marbles in the sun for a good long time.
2) Quickly put one white in one jar and all the rest in the second jar.
3) If given the full jar, the black ones should be noticeably warmer than the white ones.
Not 100%, but every little bit helps.
adj
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william wu
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Re: Easy: Marble Jars
« Reply #20 on: Jan 28th, 2003, 7:32pm » |
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Came across a cool quote by late mathematicians Jerzy Neyman and EL Scott, from probability / statistics.
"Each morning before breakfast every single one of us approaches an urn filled with white and black balls. We draw a ball. If it is white, we survive the day. If it is black, we die. The proportion of black balls in the urn is not the same for each day, but grows as we become older.... Still there are always some white balls present, and some of us continue to draw them day after day for many years."
-- J. Neyman and E.L. Scott
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| « Last Edit: Jan 28th, 2003, 7:33pm by william wu » |
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Luke
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Re: Easy: Marble Jars
« Reply #21 on: Feb 6th, 2003, 2:10am » |
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74.74% recurring?
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Short Circuit
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Here is another sure trick.
Break a small piece off 49 white marbles and put them in jar with blacks so that you can easily distinguish them the moment you pick. Leave one white of course in the jar with whites. 
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Just Thinking
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What if you take 25 white marbles out of the one container and put them in the black container. Then if you are given the white container, you can feel it is a little more empty and you know it is all white (100%). If you are given the heavier jar, then you know it has 25 white marbles and 50 black marbles, or a 1 in 3 chance (33%).
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towr
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Re: Easy: Marble Jars
« Reply #24 on: Nov 11th, 2003, 12:42am » |
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I don't think you're allowed to choose on weight, and it doesn't really help knowing which jar you're handed (unless you can somehow avoid picking a black marble from it). Regardless the first solution is better, since it also maximizes weight difference, and maximizes your chance of survival otherwise as well.
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