The three box conundrum

Im keeping this brief, (SO I DON’T HAVE TO SIT HEAR WITH THE F***ING SPELL CHEKER FOR THE NEXT FRIGING HOURE.) The problem is, ‘As Dan had already pointed out', there are 3 main approaches to solving problems involving Probability. The Classic view which was formulated by 'Laplace and De Morgan, (Do feel free to Googal).. argued that Probability refers to a state of mind, and I quote 'None of our knowledge is certain, the degree or strength of our belief as to any given proposition is its probability'. end quote. The mathematical theory (and there are too many names to mention), shows us how to measure each proposition and how each measure can be combined in a calculus. So math is only part of the solution. Another view defines probability as an unanalyzable, but having intuitively understandable, logical relation between propositions. 'Maynard Keynes'. The third view of probability is based on statistical concept of ‘relative frequency’. Developed by 'Bernhard Bolzano' (1781 - 1841) and more recently by (R. A. Fisher) As applied in many of the arguments in the latter post, statistical probability relies on the idea of the 'relative frequency', of (an event), "in a class of events", and so can’t be taken by its self, as in the cases of having ‘one’ shot, at the Key’s in the three box’s. The event has to be repeated, or it must be carey’d out on a 'class' of boxes, if you are going to apply statistical probability. Another example (as applied in the later examples) goes something like this: We have three urns A, B, C, one urn contains only black balls and the others contain white balls, you draw from the C urn, - knowing it to contain black balls, and the probability is 'demanded' that this ball is black. If you do not know which of the three urns contain black balls only, so that there is no reason to believe that it is C, rather than B or A, then each is equally possible, and since a black ball can be drawn only in the first hypothesis, the Probability of drawing it is equal to....1/3rd. If you know that the urn A contains white balls only, the probability extends only to the urns B and C and the probability that the ball drawn from the urn C, will be black is...1 half. And so on.. In other words, 'As was pointed out', there is no one way to approach this paradox and there is no one correct answer that is in any way more correct, than results produced by using any of the other approaches to probability. It depends which methodology you use to answer the question in the first places. GOOGAL ‘Blaise Pascal’ or ‘Pierre de Fermat’, Thay first pout forward Probabilities, and im sure they would have been more than happy to tell us all about it, if they were not dead.. This age old problem has been argued and fought over by Mathematician's and Philosophers, since Marquis DE Laplace first showed up. "We are still arguing", - Go figure!

'I'll put up a different version of the box problem later. Same basic problem just a different solution.' No.NOOOOOOOO don't let him call the FBI or the CIA, email Michael Moore. This man must be stopped...........hellllllllllllllppppppppppppp....arrgggggggggh.

I can see where Ely is coming from, but I'm afraid I don't follow the logic of Neil's answer. Either way, I'm afraid both answers are wrong. The odds are not equal, and neither are they three times as high if you switch. By the way there is no trickery in the wording. I worded it very deliberately so as to avoid any ambiguity. I guess I failed. NEIL SAID: >>"He informs you he is going to open one of the other two boxes which you have not chosen and which he knows to be empty." If I interpret it right, this is the key phrase.<< This phrase is important only for the reader to understand what is happening. It is no more key than any other phrase in the conundrum - ie you shouldn't read any more into it than what it is saying. ELY SAID: >>was he always going to open box B unless you picked it, or does he mean he was always going to open 'an empty box' regardless?<< It means he was always going to open an empty box. Surely there would be no point in opening Box B if it contained the keys, because then the puzzle would be no more? Hope that helps! [%sig%]

Trust you to confuse the issue even more. Actually you can keep the poxy car, there isn't room in it for my tools. I drive a Volvo estate, because there IS.

How about we line you all up against a wall? Whats the probability that one of us has a gun?

Yes i know, i've been trying to keep out of it! I drive a KEA any way, so i don't care anout the poxy car ether...

0 + 1 = 1

Okay, so when you make your original choice you have a 33.33333% chance of getting the car. Once an empty box has been removed, and you are offered the chance of changing your mind, your chances of getting the car are exactly 50%. So it would seem that by changing your mind you increase your chances from 33% to 50%. BUT...given that your mad uncle was ALWAYS going to remove an empty box, your chances of getting the car were always going to be 50%. You just didn't know it from the outset. Surely?

Aren't statistics great

I take your point Mark. It looks like it ought to be 50% because you are choosing between two boxes and one contains the keys and one doesn't. But that is not the correct answer.

Theirs a good probability that they are...

"every time we're right, you pay us a fiver, every time you're right, you pay us a favour. " and you lose what exactly? unless the 'favour' in question is helping you to get rid of five pounds.

I'm going to admit defeat here. I've tried to explain it, but it's not easy. All I seem to be doing is going round in circles, repeating myself, and inciting Flash to reach for his axe. Ultimately the only way to test this is to act out the conundrum a large number of times (with perhaps the Aston Martin being replaced by a Cadbury's chocolate button). Note how successful you are if you always stick and how successful you are if you always switch, and compare the results. Needs to be done a minimum of 30 times each way and obviously requires two people. Allow a minute for each try and you'll have the result in about an hour.

Flash! I love ya man!

*perks up at the mention of chocolate buttons* What was the question again?

To respond to this question Chose between: A B C or state weather you are 'Unseen', by selecting D

And you're Hypothesis will be correct dublin. But there are other ways to approach the problem and results produced will also be corect... 'in there Lay's the problem'. For a Cadbury's Egg, i may have a go though?

Thank fuck there weren't four boxes.

Ely - my email has suddenly invented predictive text facility for itself, which it is just getting completely wrong. I am trying to work out how to turn it off - for some reason it also wants to type the word D*O*W*N as doand, which isn't even a bloody word...

Would have made a good bonfire..

Shoot the bastard and take the car anyway. That makes it 100% certainty you have an expensive car you probably won't be able to afford to run. By the way, which model is it and what colour?

Four boxes would have made the problem clearer and the answer easier to accept.

Radio, if you vote Kerry in November, I'll agree with anything you say about these boxes. How's that for an offer.

It's the one that costs £105,000 (don't remember the model number) and bottle green.

I've decided to bet a large amount of money that Bush will get in again - as some consolation if he does: I'd rather lose!

It's maths - when you pick your first box, the odds are 1/3 that the keys are in the box, and 2/3 that it is in one of the other two boxes. As soon as one of the other boxes is removed, the odds are 1/3 that you have got the right box and 2/3 that it is in the other box. So you should always switch. It is called the Monty Haul problem and is a classic - it makes no sense to the way our minds work, as we see two boxes and think it is fifty fifty so you may as well stick. But you should always switch. When you make your decision, you KNOW that it is more likely than not that you have picked the wrong box (1/3 chance of being right 2/3 chance of being wrong) and there's no point switching at that stage because whatever box you pick has the same odds - but the odds flip dramatically once one of the two boxes you DIDN'T pick get removed. Because the odds of the keys being in one of those TWO boxes was 2 in 3. And you now have a straight choice between a 1 in 3 chance of being right by sticking to your first choice, or a 2 in 3 chance of being right by switching. Dublindian, do you know Newcombe's Paradox?

I have to say that the quality of prize is what has made this puzzle worth thinking about. Most would have offered a Ferrari or a Rolls Royce and then, well I'd have probably said: "Look, Unc! Either give me the car or don't. all this puzzle nonsense is frankly demeaning. If you're gonna pull this kind of sh1t every christmas I'd rather have a book token. Honestly! sometimes I wish they'd left you in Vietnam." But for an Aston Martin I'd be offering to take the challenge in stockings on live TV.

I don't know dude. Bush can even pronounce his own name and Kerry feels like the Scarecrow from the Wizard of Oz to me. Both are killers and I don't care to ellect a killer. Who's running that's never been involved with killing people? I don't get into politics too much. I have a difficult enough time with people ragging me for not being a religious believer ya know... On the other hand, that reason alone is enough to keep George out of office. Perhaps I should re-evaluate my priorities.

Ah the Monty Hall Dilemma. This has baffled some of the finest mathematical minds in the world yet about half the poulation (regarldless of ability) can see straight through it. Stats can be funny like that. Since there were three boxes your original choise was probably wrong, since there is only one alternative left that is probably right.

Well done Andrew and Dan, you're absolutely right. I changed it from the original (which involved goats behind doors, and American gameshow host Monty Hall) so people couldn't google the answer. Sorry Andrew, not familiar with Newcombe's paradox. I chose an Aston Martin because I recently attended a motor show in London and came face to face with one for the first time, and even though fast cars and indeed cars in general bore me rigid, I must admit it looked pretty awesome. Another way of thinking of the problem is that in effect your uncle is offering you a choice between picking one box (A), or two boxes (B & C). Obviously two boxes are going to be twice as likely to provide you with the car as one box. In order for you to take him up on the two box option all you have to do is pick C. At the risk of labouring the point, it's as though he has put boxes B and C together into a third box (which we can call D) and is saying in effect: choose between A and D. While A has a one third chance of containing the keys, D has a two thirds chance of containing them. When you choose C you are in effect opening Box D.

My head hurts.

Er..why didn't the chappie just rattle the boxes before selecting? If the Uncle was eccentric i bet there was a banana instead of car keys in the box that wasn't empty..........or a bowl of custard. And as for Newcombe despite being an aussie he was a bloody good tennis player. as for his paradox, i didn't know he had particularly famous pair of Doc Martin's.

yes it is, it's what fat people suffer from after too long in the tub

Back to the original. This is the three door problem in a chapter of 'Curious Incident....' The trouble is, it's mathematical sophistry, and those who've plumped for the 2/3, 1/3 answer have been fooled. The chances are 50/50, plain and simple. You can prove this by showing what's wrong with the probability diagram, but I'm not sure how to do it here... If I had a piece of paper, I'd draw it out to show you how the trick is done. OK, try this. All the pre-amble is unnecessary. A mathematician would discard it. Cut out the padding, and you are left with this: Two boxes. One has car keys in, one without. As your uncle is giving you the choice, you haven't actually chosen anything yet. Just take the puzzle from this point in time. Two boxes - one empty. 50/50 choice. That's it. In 'Curious Incident...' the trickery is done by showing you a probability diagram with three possibilites in it. In two of the cases, the keys are in the other box, and in one, they aren't. So you think 2 in 3 it's the other box. What they've done is left out a fourth probability line that branches off from the third option - namely, that if your original choice is the box with the keys in, your uncle can open *either* of the other two boxes. In a true probability diagram, you must follow both of these paths, even though the outcome is the same. So there are actually two ways you can pick the right box, just as there are two ways you can pick the wrong box. 50/50. Trouble is, I'm really not sure if the protagonist, or even the author of the book, are in on the trick. Andrew - the trouble with your maths is that when Box B is removed you can't add its probability to Box C anymore than you can to Box A. You have to start again, or else divide Box B's odds bwteen the two faily - so 1/6 to Box A and 1/6 to Box C. 1/6 + 1/3 = 1/2. So both Box A and Box C now have a 1/2 probability of being right. Any A Level maths student should be able to see right through it, but I guess if you're not hot on maths, it's convincing. I'm actually very impressed - I've never seen a purer example of how susceptible the human mind is to a strong internal argument, and it makes me wonder how many times I've been fooled by the same sort of thing.

Sorry Hen, but you are completely wrong on this. Firstly, this solution is borne out by experimental results. When people are confronted with the three box conundrum several times over, if they switch their choice at the end every time, they will win the car twice as many times as they would if they stuck with their original choice every time. If you don't believe me, try it with your mates. You probably need to try it about 30 times each way and I absolutely guarantee you will get very very close to the two to one result that Andrew and I predict (and therefore a long way off the 50 50 result that you predict). You may not get the exact result, because probability is like that. But the more times you repeat the experiment, the closer you will get to the two to one result.

My Head Hurts too...I like George's answer best..it gave me a good old chuckle... do you know the unexpected hanging paradox? A prisoner is told that he will be hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur before it happens. He cannot be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason, and the same argument shows that he cannot be hanged on any other day. theoretically it isn't possible to hang him unexpectedly... Nevertheless, the executioner unexpectedly arrives on some day other than Friday, surprising the prisoner.

*hangs self*

The day the Prisoner was told?

This paradox is similar to that in Robert Louis Stevenson's "bottle imp paradox," in which you are offered the opportunity to buy, for whatever price you wish, a bottle containing a genie who will fulfill your every desire. The only catch is that the bottle must thereafter be resold for a price smaller than what you paid for it, or you will be condemned to live out the rest of your days in excruciating torment. Obviously, no one would buy the bottle for 1¢ since he would have to give the bottle away, but no one would accept the bottle knowing he would be unable to get rid of it. Similarly, no one would buy it for 2¢, and so on. However, for some reasonably large amount, it will always be possible to find a next buyer, so the bottle will be bought. I admit Liana ...I don't enjoy these much either...I was subjected to them when I dabbled in philosophy, something I'll never do again!

Hen said: >>Andrew - the trouble with your maths is that when Box B is removed you can't add its probability to Box C anymore than you can to Box A. << Excuse me Hen but that's exactly what you can and should do. Let me run through the explanation very carefully again and hopefully you will see where your confusion arises. At the start of the story you pick Box A. I think you will agree there is a one in three chance that the keys are in Box A. That must mean there is a two in three chance that the keys are in Box B or Box C. Again, you can't quarrel with that. At this point, your uncle reveals that Box B is empty. But it doesn't change the probabilities, as described above. There is still a two in three chance that the keys will be found in Box B or Box C. The only thing that has altered is our knowledge of the contents of one of those boxes. We now know that Box B is empty. So the probability of keys being in Box B is zero. That means that the whole of that "two out of three" probability resides in Box C. So by picking Box C you really do have a two thirds chance of being right. Whereas with A you only have a one third chance.

I cant bear them... they seduce me because they are made of words... I start reading, and then they pop out of the Maths box and hit me with a club full of nails. Yuck.

Excuse *me*, but that's not how you do maths. Your problem is here: "We now know that Box B is empty. So the probability of keys being in Box B is zero. That means that the whole of that "two out of three" probability resides in Box C." No. That's not what it means, and it's not how you do it. You *cannot* transfer the probability from box B to C, any more than you can from B to A. You must divide it evenly between the two. Think of horses. In a single race, the probability of each horse winning has to add up to 1, because one horse must win. If you have three horses, each with an equal chance of winning, and one can't race, you have to recalculate the odds. You *don't* add the disabled horse's 1/3 onto one of the other horses. You certainly don't add it onto the horse you didn't bet on. It's a trick. It's designed to make you think it works by a trick of internal logic. But using maths, you can always find the point where it goes wrong.

*Albanian family decide to leave*

What surprises me most about this thread is how those that oppose(d)Dublindian have done so using theory. Although I understood the point about the Uncle's knowledge swaying the odds I did as Dub suggested and experimented using 3 differently coloured pieces of paper. I thought of a colour (ie the box containing the key) and Mrs. S. had to guess which one I had selected. First time, she got lucky and chose the correct colour. If she had changed her mind after I removed one of the wrong colours, she would have lost. The next 6 times, she chose wrongly and would have won the pretend car if she had changed her mind. That's better than the 2/3 chance predicted... but is flawed due to my small sample (shut it). I wanted to do 30 but Mrs. S. has a very low boredom threshold. Mykle, the first line in this thread says "This simple puzzle can provide hours of fun for all the family." Four or perhaps a hundred boxes would make it so obvious that no family fun would be had at all nor would I have been so entertained by those that have a degree in missingthebleedingobvious! smileyface.

OK if you really believe all that guff, take up my challenge. Do the three box conundrum with your mates say 30 times each way and you will discover where the truth lies. Your reluctance to accept the result is, I have to say, perfectly understandable. When this problem was first published in an American magazine, there were professors of mathematics up and down the USA who challenged the solution, saying exactly the kind of thing you have been saying. In the end they all backed down when they saw the error in their thinking. OK let me make one last attempt to explain it to you. Here's another way of thinking about the problem, which I sort of touched on in an earlier post. Imagine your uncle invites you to choose one of the three boxes and you pick A. Imagine that he then gathers up the unpicked boxes (B and C) and places them in a fourth box which he calls D. He now says you can stick with A or you can choose D. In effect he is saying you can: * stick with the 'one in three' option (A) or * switch to the 'two in three' option (B + C, now known as D) Remember we are talking about an Aston Martin here. Common sense dictates you must go for the 'two in three' option. Anything else would be stupid. Well, guess what, this is in effect what your uncle is offering you when he invites you to switch your choice and reveals to you that B is empty. When he reveals B is empty, he is in effect making it possible for you to open that hypothetical box D - simply by choosing C. Think about it Hen. You know it makes sense.

Jude with the hanging paradox, the only reason that the days are eliminated as possibilities is because he knows that IF he is still alive on the day before then it HAS to be the next day If the hangman turns up on Monday and hangs him then he had no way of knowing it was going to be that day.

oh, and the imp in the bottle: does the imp stay in the bottle when it is sold and is the deal about it having to be sold on for less money also passed on. PLUS, having read your post I don't see the paradox, you conclude that if someone sets a reasonably high price then they will sell the bottle no problem so where's the mystery?

right, I've been to the maths website from where the above examples have been taken and looked throuigh the others there. all very interesting but there's one I really don't get. It reads thus: "You buy 100 pounds of potatoes and are told that they are 99% water. After leaving them outside, you discover that they are now 98% water. The weight of the dehydrated potatoes is then a surprising 50 pounds! " now I'm a clever guy and this is obviously very simple or there'd be more of an explanation but I don't get it help, anyone?

I'm a convert you got me If you take it to 100 boxes and the uncle removes 98 then you can see that it's his knowledge of where the keys are that increases the odds: "Right I've removed all the boxes that I know it's NOT in, so it's either in the one you picked (unlikely) or the one I've left" I also tried this on my other half and she made a very interesting point. If it were to actually happen then the statistics would go out of the window and the motivation would take over. If you suspected you uncle would rather you didn't win the car and was therefore trying to get you to pick a loser (chase the lady stylee) then his "would you like to switch" would seem suspicious and you'd probably stick to your guns. If you suspected he was trying to give you a gift then it would seem like "Oh, you've actually picked the wrong box so here's a second chance dearest nephew" so in reality I believe that we would work out our 'chances' based more on the situation than the maths

Yes Liana, that's exactly it. The words are a con. My brain refuses to look at these things. It just cannot pull all the requisite strings out of the tangle and thread them together to make sense. It makes me want to run out of the classroom and hide behind the bikesheds.