The three box conundrum

Dublin: Just as a matter of interest to me.... if no-one knew which box the key was in, but once a box was chosen if the uncle said "would you swap what's in that box for what's in these two?" - would the odds be identical to the original problem?

Jesus christ... are my posts invisible? Look, forget all this theory bollox, go away and try it out like I did. Postulating ninnies.

Doug Hofstadter did that at a conference and got as high as fifteen dollars, because the bidders didn't realise until they were drawn into it that the loser would lose money, and face, and the dollar bill. Ely - yes, you know the rules that I have set out above. The criticial thing is that he predicts and places the money BEFORE you open the box. So, the money is in those boxes, no matter what. This is relied on for one camp, who say 'no matter what the prediction, you make more money by opening both boxes, since you get either £20,000 plus nothing, or £20,000 plus a million. If a friend could see into both boxes, he would ALWAYS want you to take both... The alternate camp say, but the only way to stand a good chance of getting the million is to decide to open Box B only, because the money is only in there if he thinks you will open Box B, and that therefore you will lose the money if you try to be greedy... Personally, I see the logic and force of Both Boxes, but I would always only open Box B - if there was nothing in it, I'd want it to be because the being got me wrong, not because I was greedy and thus got what I deserved. Bottle imp - it is not recovering your investment that is the concern, it is being able to pass on the hot potato - for it to work, you would have to explain the terms to the person you were selling it to and have them understand it. The advantage of paying a lot is that you increase the number of steps between the person you sell it to and the person at the end of the chain who is going to get caught with the torment - ten steps means that someone five steps down might say 'no thanks, too big a risk' whereas 20,000 steps is too distant for anyone to worry about - so probably £200 is as much as you would need to pay.

The odds are not reset because you do not start afresh, your uncle only opens the box after you have chosen one and he knows which you have chosen. He is in fact reacting to your initial choice. I'm gonna try a different way of explaining this. Let us imagine two parrallel universes, we shall call the Universe 1, and Universe A In Universe 1 you have chosen the correct box. In Universe A you have chosen one of the wrong boxes. I think we all agree that (without cheating) you are more likely to end up in Universe A. Your uncle then reveals one of the unchosen boxes as empty and offers you the chance to switch. The you in Universe 1 should stick, you already have your hand on the correct box, what a shame you do not know this. The you in Universe A should switch, there is only one box left so it must be the correct one, fortunately after reading this thread you know this very well and do so. Note that *nothing* random happons at all after you chose your first box so no more universes are created. The probability of it being the right box DOES NOT CHANGE. Or another way of looking at it. Your uncle lets you choose a box (1/3 likely to be right)and then tells you that one of the other boxes is definitely empty but doesn't reveal which one. Does this make the box you chose now 1/2 likely to be right? He then reveals which one. Does this increase the odds of you being right? Or yet another way. After choosing a box your uncle says you can switch to one of the other two. "If you decide to switch." He says. "I'll even open one of the two I know to be empty. But you have to decide to switch first." Obviously you switch. How is this different?

yep that makes sense too. How come everyone who explains it I believe, even when they differ? this is why I write. I just don't get maths at all. You are all so damned convincing. I was on t'other side till I spoke to 'maths' friend. Will have a word with him tomorrow - and maybe get him to try it out with me. Your posts weren't invisible Stormy - nobody would play that's all.

I believe they would Mykle. By switching to the other two boxes you would be doubling your odds of winning, as compared to sticking with just the one box.

Wrong! The odds are exactly the same. No matter what you do. They won't change. If you choose 1 box, there is always another empty box. Revealing it doesn't change the odds of you having selected it. 1 in 3, every time, no matter what.

Just read it Andrew. (Very Clever) The Classical ve the Quantum! Relativity ve Probability, or free will ve determinism. Which are you? I like this one..

That's just your opinion. One I wouldn't want to risk my life on at that.

So, surprisingly, the only need for the uncle to know which box the key is in is so that he can be certain of opening an empty box before he offers the swap. I must say, Dublin, that this doesn't seem logical as I assumed that it was the uncle's knowledge of where the key was that changed the odds - when, in fact, it's just essentially a misdirection.

How can you double your chances by switching from the original choice to one of the other two when you still have to make a single choice? My simple stupidity (a characteristic that appears to give the twat much pleasure and satisfaction), tells me that a choice from 3 gives a 33.3% chance of getting it right, and a choice from 2 gives a 50% chance of getting it right. It appears that the chances of getting it right are no different, because at least one of the other 2 boxes was always going to be empty. Opening 'one' that 'he' knows is empty means absolutely nothing, the other one may be empty also so it makes no difference at that point whether you change or not. The odds of being right out of two is obviously better than being out of three, BUT if the odds are 50% after he's opened 1 empty box, then it's STILL 50% whether you pick one of the 2 or the other. Is this really about mathematics, or is simple logic enough to explain the odds?

You guys are surely winding me up. I've just shown that switching gives you double the chance. It's no use saying you don't agree. You have to tell me which part of the above explanation is wrong.

Dan, bollocks. YOU don't start again, but anyone that changes the rules halfway through with ME, is gonna get a fresh deal.

so you know that if he thinks you're gonna play safe then he'll punish you but if he thinks you're a risk taker then he'll rewarde you with a million. Any way you look at it you can be seen as being greedy and, therefore, deserving of punishment. both boxes means you may be hoping for the double whammy, just box B means a guaranteed 20k isn't enough for you, either way you're greedy in someone's eyes. I'd always go for box B because if he was so wise then he'd know I would do that and reward me, if it's empty I can always say, "Not smart after all are we?"

Hi Dan. That was just me signing of having spent most of the time quietly following this debate. 'with grate interest i may add'.

*chops up boxes for firewood*

Personally I value £1000 way above a short lived sense of self satisfaction.

'Marquis De Laplace' say you cant chop up box's' so pout them back! Places..

*looks at john and drools menacingly*

I admire your persistence dub.

Doesn't really matter now Flash, some thieving B. has taken advantage of all this hesitation over which box to open to boost the car!

I think I now see where Mississippi's confusion comes from. And OK it's probably my fault. In my original post setting out the problem, it was (in my view) clearly implied, but not expressly stated that after you choose box A, there are two possible scenarios for your uncle. There is the first possibility (which is the example given) that if the keys are in box C, your uncle will have to open B to reveal it is empty. But there is also the possibility that: if the keys are in B, your uncle will have to open C to reveal it is empty. It is this extra possibility that increases the odds for you if you switch. The point is that when you stick with A there is only one one-in-three possibility of your being right. When you switch to the unpicked box (in my example this was Box C), there are two one-in-three possibilities of your being right (either the keys are in B and there are no keys in C or the keys are in C and there are no keys in B). To make explanations simple I always talked about B as empty and C as being the remaining unpicked box. But it could have been the other way round.

hehehe.. Now what was the probability of that happening? *Drools back at Flashy*

Bearing in mind what I have just said, this is the clincher. (Once again it is understood, but not expressly stated in my oiriginal post, that uncle makes his selection completely at random. He could place the keys in either A, B or C) Right having got that out the way here is the explanation: You have to agree there are three possible locations for the keys: Box A, Box B, Box C. Each of these has a one in three chance of being the location uncle chooses for the keys. Therefore, if you select A, and stick with A, you are going to be correct one in three times. If you select A and then decide to switch, you can either switch to C (if uncle has revealed B as empty) or switch to B (if uncle has revealed C as empty). Both of those possibilities have an equal one-in-three chance of happening. But there are two of them. Two one-in-three chances add up to two in three. It's as simple as that. I think if you trawl back through the posts someone else makes this point (Dan or Andrew, can't remember). Right I'm off.

*weeps softly*

Think I'll kill myself.

*weeps loudly*

Ely,Missi, John and Dublin too should all have better things to do 200 posts nearly on silly boxes i wished they'd go and play with their...

I was thinking the same, Liana, till the other threads picked up a bit...hehehe.

£20,000 sounds great to me.

" I understand that time is at a premium for you, but it smacks of cowardice to run away when you don't get the response you hope for." You misunderstand me. I'm trying not to spend so much time here because I'm not very good at keeping my cool, and just taking it for what it is. And they *are* right about this. I was roundly, soundly defeated.

Your explanation leaves out one thing. Switching provides no better odds. Reason being, you don't improve your odds of being correct to 50/50. What you do is change the odds of selecting a wrong box on the switch to 50/50, but this has no effect on the origional problem. This is a clasic bait and switch logic puzzle. Throw in something meaningless and let the brainies bend off on a tangent with calculations when the actual problem solution doesn't change one bit. When you switch, you also have a 50/50 chance that you're deselecting a correct first guess, which nulifies the 50 percent chance you get it right when you switch. Bottom line, the odds are 1 in 3 you ultimately end up with the correct box, no matter how you size it up. Can't explain it any clearer. Pure statistics, combination theory of games. Business math 101, community college level.

God this is probably the best case of ambiguity aversion that i have ever seen...Jesus!

Mississippi said: <> The only chances you are being offered in this puzzle are: A one in three chance (33.3%) if you choose box A A two in three chance (66.6%) if you choose boxes B & C at the same time. The key to understanding this is recognising that what your uncle is offering you is a choice between those two options. OK the puzzle is packaged in a way that makes that less than obvious, but essentially that is the choice your uncle is offering.

Mykle said: <> I'm not sure I understand why the uncle's knowledge is such an issue. As you rightly say he needs to know where the key is so he can reveal the empty box. It is possible to imagine a scenario where he didn't know where the key was and just took a blind guess at opening one of the other boxes. If by chance he happened to pick the box with the key, he would have blown the puzzle. If by chance he picked the empty box, then the odds would be exactly the same as if he had known all along. So in that sense I guess the uncle not knowing does not affect the odds.

Sorry Dublin, but in your opening post where you laid out the problem you specifically said, and I quote, >> He invites you to choose one of the boxes << I insist that ONE means ONE, that means initially you have a single choice from 3, ie. 33.3% of choosing correctly and eventually having eliminated one box, a choice from 2, ie. a 50% chance of being correct. At no point does the original wording suggest your above statement to be what is actually on offer, >> A two in three chance (66.6%) if you choose boxes B & C at the same time << It's the 'at the same time' bit that is the problem. Does this mean you can open BOTH of them? If so, I agree it doubles your chances of success. Perhaps it's your wording of the problem that is at fault and not my logic. I admit my maths is crap.

I bet you're voting Bush rather than Kerry or Nader, RD.

Now, there's a jump of logic. Personally, I'd rather see Nader in. I don't vote, I refuse to take part in controversy. I rather stand back and detract

"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. If the car keys are in Box C, he then HAS to open Box B as that is the only box which you have not chosen and which he knows to be empty. If the car keys are in Box A, he had the choice of opening either Box B OR Box C. There is only a 50% chance of a 50% chance (therefore 25%) that he would open box C and reveal it to be empty. He didn't, therefore there is a 75% chance that box C holds the keys. Therefore the answer is e) , your odds are three times as high if you switch.

So you have three boxes. Which one do you put your cross in?

Is that an Ink Pen or a crucifix?

OK, I was wrong about your voting intentions, but not about this problem. To be honest I do not understand your counter argument, and you seem unable to accept my explanation so we've reached stalemate. We've probably exhausted all the possible things to be said on this, so we'll just have to agree to disagree. Nice talking to you. I too like Nader, but strikes me a vote for him rather than Kerry is a vote for Bush.

*sharpens axe*

Here's an idea - all those lining up against Dublindian and me, let's play for money. We'll run the test a hundred times and Dub and I will always play switch - every time we're right, you pay us a fiver, every time you're right, you pay us a favour. If the 50-50 theory is correct, we should end up even. If, however, Dub and I are right, then Hen and Missi should end up paying us around £325 and we pay them back £175 making us £150 better off. Screw all the maths and probability and arguments - I don't care if you don't think I'm right, provided you're prepared to play me at this game for money... By the way, I also know a good bet about whether in group of 30 people there are likely to be two people who share a birthday...what do you reckon the odds are? The common sense part of the brain says, 12 months in the year, all with around 30 days so 12 times as many birthdays as there are people, so around 1 in 12 . The maths say that it is actually about 1 in 2.

Well Hen not many people (myself included) have the courage to say those last five words, so all credit to you mate.

"He adds that you should read nothing into this offer. He was going to give you this choice whether or not you selected the right box in the first place." what do you mean by 'this choice'? 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? Other than that, I can see no riddle here. you only have two pieces of information. 1: it's not in box b 2: it IS in one of the remaining boxes your chances are equal therefore, unless I'm being thick and missing some huge wet kipper in the face of a clue somewhere.

Hmm... Jon came round last night with his head a-pounding over this riddle-me-ree. Statistics have no actually existence in the world - imagine this scenario: Nephew A is given a choice of three boxes, and picks one. Uncle then removes one of the remaining boxes and invites Nephew A to choose a box. According to the conundrum, Nephew A, is faced with a 66.6% probability versus a 33.3% probability. However, if at this point, a second child, Nephew B, enters the room, and *unaware of the removal of the previous box* is asked to pick, he has a fifty-fifty chance of choosing the correct one. The thing that throws people is when they forget that probability is all about predicting things. There isn't two thirds of a key in one box, and a third in the other. There is one key in one box, and no key in the other two. If the Uncle simply showed Nephew A which box contained the key, the boy would know with 100% certainty which box contained the key. His uniniated sibling, Nephew B, would still only have a one in three chance. So neeyah. All this proves is that statistics is a flawed, provisional way of describing the world.

it's okay, you can be wrong on both accounts, doesn't bother me hehehehe... That's why I don't vote. No vote isn't a vote for Bush. I have a Chinese lady in my office that tells me in China they call Bush, Shrub. You know...a little Bush..His dad...he's the little one... hardee har harrrr... I'll put up a different version of the box problem later. Same basic problem just a different solution.

*Quantifies the probability of blood shed*

I know nothing about maths but wasn't rad agreeing with you? no? actually, don't answer this time. I'm too thick to know. *fergal collapses in maths induced coma*