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Title: Easy: Marble Jars Post by ootte on Jul 24th, 2002, 12:54am 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. -- Oliver |
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Title: Re: Easy: Marble Jars Post by James G on Jul 24th, 2002, 1:04am 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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Title: Re: Easy: Marble Jars Post by ootte on Jul 24th, 2002, 1:10am on 07/24/02 at 01:04:21, James G wrote:
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 ... -- Oliver |
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Title: Re: Easy: Marble Jars Post by Gabin Kattukaran on Jul 24th, 2002, 2:49am 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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Title: Re: Easy: Marble Jars Post by Josh on Jul 25th, 2002, 3:32pm on 07/24/02 at 01:04:21, James G 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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Title: Re: Easy: Marble Jars Post by Alex on Jul 25th, 2002, 4:50pm Even better: Just keep redistributing marbles for all eternity. It specifically states there is no time limit. :) |
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Title: Re: Easy: Marble Jars Post by Ion Rush on Jul 27th, 2002, 3:37am 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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Title: True, the guards could shake Re: Easy: Marble Jars Post by James G on Jul 31st, 2002, 12:43am 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. :P 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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Title: Re: Easy: Marble Jars Post by Brion on Jul 31st, 2002, 3:02pm 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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Title: Re: Easy: Marble Jars Post by Donkeyboy on Jul 31st, 2002, 10:59pm 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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Title: Re: Easy: Marble Jars Post by william wu on Jul 31st, 2002, 11:12pm 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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Title: Re: Easy: Marble Jars Post by Donkeyboy on Aug 1st, 2002, 1:02am aaaaahhhhhh :) You're right.....I was assuming that marbles needed to be swapped. Thanks for clearing that up :) |
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Title: Re: Easy: Marble Jars Post by James Paige on Aug 8th, 2002, 11:18pm 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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Title: Re: Easy: Marble Jars Post by James Paige on Aug 8th, 2002, 11:19pm "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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Title: Re: Easy: Marble Jars Post by James Paige on Aug 8th, 2002, 11:21pm 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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Title: Re: Easy: Marble Jars Post by James Paige on Aug 8th, 2002, 11:22pm 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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Title: Re: Easy: Marble Jars Post by jeremiahsmith on Aug 9th, 2002, 12:47am Good story. :D |
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Title: Re: Easy: Marble Jars Post by Mongolian_Beef on Aug 13th, 2002, 8:24pm 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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Title: Re: Easy: Marble Jars Post by Gleb on Aug 16th, 2002, 5:08am on 07/25/02 at 15:32:35, Josh wrote:
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Title: Re: Easy: Marble Jars Post by avjewe on Oct 2nd, 2002, 6:22pm 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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Title: Re: Easy: Marble Jars Post by william wu on Jan 28th, 2003, 7:32pm 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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Title: Re: Easy: Marble Jars Post by Luke on Feb 6th, 2003, 2:10am 74.74% recurring? |
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Title: Re: Easy: Marble Jars Post by Short Circuit on Nov 7th, 2003, 2:36pm 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. ;D |
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Title: Re: Easy: Marble Jars Post by Just Thinking on Nov 11th, 2003, 12:15am 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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Title: Re: Easy: Marble Jars Post by towr on Nov 11th, 2003, 12:42am 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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Title: Re: Easy: Marble Jars Post by PermanentInk on May 27th, 2004, 1:24am Can one put all the marbles in one jar? If handed the empty jar I'd be unable to follow the instructions to pick one marble, but neglecting that, there's a way to interpret the problem that gives exactly 75% odds of living. It says that if I draw a white I live, and if I draw a black I die. What happens if I draw no marble, because there are none? Since it doesn't explicitly state that I die, I assume I live by default. It may seem a bit of a stretch to simply neglect the requirement that I draw a marble. But note that the problem states very explicitly that the *only requirement* is that every marble must be in some jar. It does not say that each jar must contain at least one marble. |
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Title: Re: Easy: Marble Jars Post by grimbal on May 27th, 2004, 4:59pm on 10/02/02 at 18:22:46, avjewe wrote:
I wouldn't put the white marbles in the sun. Make them wet and put them in the shade. Use the mini-fridge if there is one in your cell. |
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Title: Re: Easy: Marble Jars Post by Tommy Vercetti on Jun 21st, 2004, 6:26pm noooooooo people if u put one white marble and 49 black marbles in a jar that doesn't make it 50% its more like a 2% chance of winning where did u all learn math? |
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Title: Re: Easy: Marble Jars Post by BNC on Jun 21st, 2004, 9:42pm on 06/21/04 at 18:26:03, Tommy Vercetti wrote:
And where did you learn to read? ::) Read again through the thread -- it's not long. |
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Title: Re: Easy: Marble Jars Post by Marble Madness on Jul 19th, 2004, 9:53pm I could be wrong, but doesn't it say that after redistribution that all marbles must be in ONE jar? |
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Title: Re: Easy: Marble Jars Post by towr on Jul 20th, 2004, 12:27am on 07/19/04 at 21:53:08, Marble Madness wrote:
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Title: Re: Easy: Marble Jars Post by WombatDeath on Jul 20th, 2004, 2:57pm on 06/21/04 at 18:26:03, Tommy Vercetti wrote:
Can't speak for anyone else but I learned at Honest Jim's Temple of Infinite Cosmic Enlightenment. Numbers 1 to 9 are free, but it's $50000 (cash) to learn about the zero. Course, you don't know what the hell that means until after the lesson. And, coincidentally, the Temple is registered at the same address as Honest Jim's College of Emphatic Debt Collection and Applied Ultraviolence. |
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Title: Re: Easy: Marble Jars Post by Craig on Aug 18th, 2004, 6:38pm Wouldn't your best odds come from putting 1 white marble in one jar and the rest of the marbles in the other jar? That way you have a 50/50 chance of getting the jar with one marble and you win. But if you get the other jar you still have a 49:99 chance of winning. I don't know probability but would that give you about a 75% chance of living? -Craig |
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Title: Re: Easy: Marble Jars Post by william wu on Aug 19th, 2004, 5:43am Yes Craig, that is correct. Actually, it has been mentioned several times in this thread already (see posts above you and on previous page). |
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Title: Re: Easy: Marble Jars Post by Guest on Aug 22nd, 2004, 6:09pm Definately, put 1 white marble in it's own jar , put all other marbles in the other jar. As for putting the marbles in the shade, etc. etc., the trick would be to count every single marble, and then palm a white marble, but dont let them know, and then complain that theres only 49 white marbles. They put another white marble in (hopefully), all better. Because now you have 51 white marbles. ;) (I'm loving the cowboy hats!!) Now put 25 marbles of each colour in the two jars. (Trust me here0 Now, when you pick a marble, take your palmed white marble, put your hand in a jar, and pull your hand out, clenching your 51st white marble, the one you palmed earlier. Sorted! Well .. I thought it was a good idea... |
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Title: Re: Easy: Marble Jars Post by rmsgrey on Aug 23rd, 2004, 6:25am If you're planning to cheat by palming a white marble, you're better off only putting 99 marbles between the two jars and hoping no-one notices before you can reveal the 100th than having 101 marbles around afterwards having drawn attention to the fact there's something dodgy about the number of white marbles - whichever approach you take, you're screwed if they actually check the marbles, so you're best off not drawing their attention to the possibility... |
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Title: Re: Easy: Marble Jars Post by sachin kumar on Oct 13th, 2004, 2:06pm http://URL As per my opinion. Keep 1 white Marbels in one jar and transfer 49 into the second jar. this make the probability of surviving = 0.5*1+0.5*(49/99)=0.75(approx). this the best bet as i can say. |
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Title: Re: Easy: Marble Jars Post by Dust Mite on Oct 18th, 2004, 4:44pm I was thinking along the lines that if each jar was filled to the top(which resulted in the problem them stating that you must trade 1 for 1 in the jars and leave none out) then if each jar had all whites at the the and it were shaken the marbles will not move(given there is no room to...). |
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Title: Re: Easy: Marble Jars Post by kroml8r on Nov 7th, 2004, 5:32pm Assuming that we have put one white in one jar and the otheres a re mixed in the other jar. According to Bayes formula for total probability, Let A = event that the jar with one white marbel is picked. Let B = event that the mixed jar is picked Let C = event that a white marbel is picked from any jar P(C) = [P(C|A) x P(A)] + [P(C|B) x P(B)] P(C) = [1/2 x 1] + [1/2 x 24/49] P(C) = 0.7449 or about 74.5% Personally I'd just run away! |
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Title: Re: Easy: Marble Jars Post by kroml8r on Nov 7th, 2004, 5:41pm Sorry I should have used 49/99 in place of 24/49 which alters the answer to 0.74747474... as mentioned in an earlier post. :-/ |
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Title: Re: Easy: Marble Jars Post by Brad Sampson on Dec 6th, 2004, 3:45pm Some of you guys are crazy... Anyway, there's lots of answers. The answer with pure logic is 1 white for one jar and 49 white 50 black in the other. Of course, there's the fact that kidnappers would give you the worst possibility.(Awesome story, by the way!!!) You can alaways say to heat them up, and to chip peices off, and to palm a marble, but these are riddles. Not science problems. Although those are fun and interesting. Anyway, if I was abducted by kidnappers, I would still do 1 white and then 49 white and 50 black, because the chance of living is only 0.05050505 repeating less than 25 black and 25 white. Whereas, they might be slightly honest(not likely) or just awed by your smartness, so I think that the small chance of having death more that 50% doesn't really matter. |
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Title: Re: Easy: Marble Jars Post by GeminiWarith on Nov 25th, 2005, 9:04pm Put all the marbles in 1 jar and then grab a handful when the time comes. If you get one of each at least they can't let you live or kill you, unless the riddle says to pick one. |
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