Monty Python Problem
A host places 100 boxes in front of you, one of which contains the holy grail.
You pick box 3. The host removes all the boxes except 3 and 42, stating that the holy grail is in one of these boxes.
What is the probability that the holy grail is in box 42?
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7 solutions
I thought it would have been 50%. The other boxes were empty, and the holy grail could be in either box 3 or box 42. 50/50 chance.
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It is 50%
Of course the probability resets. It wouldn't reset if you didn't know the boxes were empty. Now that 98 were removed, and you are choosing again, the probability has to reset. Now you have a choice between 2 boxes, one has it, and one doesn't. That's a 50/50 chance and the answer is 50%.
That is not right, because originally there are 100 boxes. If you choose you box 3 it is 1 percent. Now the rest of the boxes combine is 99 percent. And then the Bridge keeper took away all the boxes except for box 3 and box 42. Now you know the boxes that are taken away are empty. BUT the probability doesn't reset. Box 3 still is 1 percent. And now box 42 is 99 percent because the probability doesn't change. Now of course you would switch to box 42 rather than stick because 99 percent is much more greater than 1 percent.
The events are related so the probability doesn't reset but does a funny concentrating thing because of the revealed empty boxes
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Even if the events are related I don't see how that makes it so. I assumed that the question was asking what the probability is AFTER the empty boxes are taken away - prior knowledge or not, when you are presented with two boxes, one of which is containing the grail, you have a 50% chance of picking the right one.
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@Madison Reardon – That logic isn't quite right. I think the bigger the number is, the easier it is to see why the other box has a much higher chance of being the box with the object in it. Imagine that instead there are a million boxes. The chance of you picking the correct box on your first guess is one in a million, which is extremely unlikely. After removing all of the other options except one, the probability that you chose the correct box right off the bat is still one in a million. Therefore, changing your guess is much more sound probability wise. There's a 99.9999% chance in this situation that you picked a box with nothing in it as your first chance, therefore there's a 99.9999% chance you'd get the box with the grail in it by switching.
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@Joe Dain – Thank you for your explanation. It is easier to understand.
@Joe Dain – yeah its even harder to see when it is 3 choices like in the original monty hall problem
@Joe Dain – Nailed it. This is a really good piece of perspective most people don't consider.
@Joe Dain – i dont really agree with any of you. this isn't just math, its logic. theres more than just simple probability. first of all someone is asking you to choose. and they are manipulating their question by changing the equation. my instinct would be there's human deception at play. taking out the grails to destroy any grasp you might of had on the concept depicted. and to say I should look at this as simple math is just a toss up of perspective. to choose box three is to follow the path set forth by another. the ladder choice (42) Is to forge your own path and take a necessary risk for such a fitting reward. and as such there is no real truth. only perspective. but like everything this is a problem created by one credited to another, to increase comprehension of many. None the less, I have enjoyed the mental debate this has given me.
I agree with you¡¡
Right, but wouldn't the odds change? If 98 of the boxes are empty, and your choices are now between two boxes, how isn't it a 50/50 chance?
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If there were 100 boxes, then 98 empty boxes were taken away and then you chose one, there would indeed be a 50% chance each way. But you chose 1 out of 100 possible boxes before 98 empty boxes were removed. Think of it as these 2 possibilities:
The holy grail is either:
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in the box you chose first time ( P = 1 0 0 1 ), or
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in one of the 99 other boxes ( P = 1 0 0 9 9 ).
When the 98 empty boxes are removed, there is no further uncertainty: if option 1, the holy grail is in the box you chose; if option 2, it is in the other box. So, in this question, P ( box 42 ) = 1 0 0 9 9 .
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Yeah, but you are given another chance to choose. It's either in box 3 or box 42. Your odds increase to 1/2 from 1/100.
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@Sha Un – At first, I thought it was 50/50 chance as well, as I had commented earlier. But I think I get it now.
At first, your odds are:
Box 3: 1/100
Other boxes (including box 42): 99/100
Then the Bridgekeeper takes away all other boxes (we know they are empty) except Box 42. The probability STAYS THE SAME, as the Holy Grail hasn't shifted boxes:
Box 3: (it stays the same) 1/100
Other boxes (minus the empty ones)(aka Box 42): 99/100
Therefore 99% chance it is in Box 42. So your odds don't increase, the chances that it was in Box 3 still is 1/100 as the Holy Grail was still put in 1 out of 100 boxes - the probability doesn't reset.
or....
Think of it this way...
The Holy Grail was put in one box among 99 other boxes. The boxes were then shuffled around like a pack of cards. Then you pick a box and put it aside. And all the other boxes go to 99 other people. The probability that you have it is 1/100. The probability that one of the others have it is 99/100 (as there are 99 other people+boxes). Now suppose 98 out the other 99 people open their boxes, only to find it empty. So now the Holy Grail is in either your box or that one other person's box. The chance that you picked the right box on your first go (no offence) is still only 1/100, as you kept your box and that 'other person' still remains part of the 'other 99' people/boxes. The chance that that other person has it is therefore still 99/100, whereas yours is still 1/100. (You and the other person didn't put it down, shuffle it about, and then choose again. If you did, your chances of picking the right box then is 1/2).
I hope you understand it now :)
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@Radhika Saithree – This didn't make sense to me at first, but now it does. Now I only have one question. are the probabilities based on a frame of reference? for you, the probability that you picked the holy grail right off the bat is 1/100 so that means that with every person that opened the box, the probability that the person with box 42 has the grail increases, as he ALSO only had one box, and the probability of him having the grail right off the bat is 1/100 as well. but as every other box proves to be empty, his probability increases until it reaches the inevitable probability of 99/100. now, could the same be said from his perspective? because as I see it, in his frame of reference, HE only has a 1/100 chance to have the grail, and as all other boxes again prove to be empty, your probability increases to 99/100 from his point of view...
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@Keenan Orth – That's true, unless you consider him to be collective with the other 98 boxes/peoples. If you consider him to be part of the other 98 boxes, then his chances stay the same. If you consider him an individual, then his chances are the same as mine. So yes, I guess it is based on frame of reference. But in the original problem, it's you and your box versus all the other boxes. Not you and your box versus 99 other people and their boxes.
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@Radhika Saithree – I see what you mean, I only meant to point out that probability might be relative, and if that is the case, than there would be no use in switching boxes. From your point of view, box 42 has a 99% chance of containing the grail, switching would be a no-brainer. But from the "view" of box 42, you are the one that has a 99% chance containing the grail in box 3. What would switching boxes then do? Which point of view is right? if you switch boxes, do you really have a better chance than if you hadn't? Which box actually has the better chance?
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@Keenan Orth – I see what you're talking about. Actually, from the "view" of box 42, "box 42" and "his companion boxes" represented the 99/100; then box 42 should rest assured that he has the upper hand (99% chance). Box 42, or the person who has it, shouldn't think that I have 99% chance of having the holy grail, because I chose one box and one box only out of the hundred options, thus giving me a 1% chance. Box 42 WAS part of the other 99% (99 boxes), but despite that the number of boxes on his side reduced from 99 to 1 (him), his chances are the same (i.e. 99%).
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@Radhika Saithree – I get it now. Box 42 is part of a collective. box 3 should be the only box that is singled out, as it is not apart of a collective of other boxes. I realize now that this is exactly what you have been saying, and I refused to look at it like that because the idea of probability being relative intrigued me so much.
@Radhika Saithree – If opening the 98 boxes was random I'd agree with you, but the fact that the person that eliminates the 98 boxes knows exactly where the grail is, this leads to a 50/50 chances.
See the problem in the other way, you are allowed to select 99 boxes, 99/100. Then, the bridge keeper takes 98 empty boxes that you selected. Do you still think you have 99% chances of win if no shift?
The first pick is just part of the game, the real pick comes when you only have 2 boxes to choose.
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@Carlos Francisco Montoya Mejía – I agree that if there were a hundred boxes and 98 removed, out of the 2 left, there would be a 50/50 chance of picking the right one. But our choice of box is not out of the 2 left, it is out of the 100 boxes altogether.
Try looking at it this way: There are 100 boxes. You choose one and put that to the left and the other 99 boxes go to the right.
Now, the chances it is on the left is 1/100 whereas the chances that it is on the right is 99/100.
Now obviously you'd want to go for the boxes on the right because there's a 99% chance it's in there somewhere, but you can't do that because because you are only allowed to open one box.
But, there's a lifeline option. That means that someone comes and takes the 98 empty boxes out of the 99 on the right, so there's only one on each side now. Your chances are same, 1% chance that it is in the box that you picked at first on the left, or the 99% chance that it "is somewhere in there" on the right.
Now, you finally have to make your decision. The left or the right? Obviously you'd want to go for the right, because there's a 99% chance that it "is in there somewhere". BUT this time, you CAN choose this option because it only involves opening one box! How lucky is that!
So the first pick is not just "part of the game", it is part of the real pick, but the only difference is that the second time of decision, your odds haven't changed, but your options have (because the 2nd time, you CAN choose to go to the right)
Hope it's helped. It sure has been amusing creating stories and characters for this "game of choosing boxes" ;)
@Sha Un – Try drawing a probability tree. First stage: holy grail in box 3 / holy grail not in box 3. Second stage (after 98 empty boxes removed): holy grail in box 3 / holy grail in box 42. You should find that your probabilities for the first stage are 1 0 0 1 and 1 0 0 9 9 respectively, and for the second stage are 1 and 0 depending on the branch from which they follow on.
Just because there are 2 options doesn't mean they are equally likely, even though you are supposed to think that they are!
I don't understand how you are all getting 99/100 as the chance for you to get the grail in a different box. Clearly the chances of the grail being in the first box are 1/100 and the chance that the grail is in any one of the other boxes is 99/100, but when you consider that the entire goal of the Bridgekeeper is to confuse and cause doubt, therefore logically one would omit the previous probability in order to show a resulting probability of 50%. You cant use the idea of Bayes' Theorem in a logical conundrum. I reference the game show "Deal or No Deal" to portray this idea in the most realistic way possible.
I've always struggled with these types of problems (hence, the English degree) until I re-read Martin's and Radhika's answers several times. And then I saw it. If I choose box 3 and the other 99 boxes are put in a pile and I am allowed to keep box 3 or trade it in for the other 99 boxes, I don't believe that would be a difficult decision: 1/100 v. 99/100. My method of envisioning the answer just requires ME to open 98 empty boxes first before getting the Holy Grail box.
I think it's the "taking away 98 boxes" that makes people want to turn it into a 50/50 proposition.
In order for it to be 99%, that would require all of the odds from the 98 removed boxes to be enclosed in 42, as if instead of removing the 98 boxes, you enclose them into box 42. Therefore it's 50%
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I see what you mean, but the thing is, in a way, the odds from the other 98 boxes all gets dumped onto box 42, because it is part of the collective other. So it's still 1 box (1% chance) against prev:99, now:1 (same chance: 99%)
It should read, "Then the Bridgekeeper takes away 98 of the boxes which he tells you he knows are empty, but does not tell you if box 3 or 42 is the other empty box." - why?
If the Bridgekeeper knows that the boxes he removed were empty because he knows which box the holy grail is in, then the probability for box 42 containing the grail is 99/100
If the Bridgekeeper does not know which box the grail is in and removed 98 boxes by luck (we only know from the semantics that the writer knows the boxes are empty, but with regards to the bridgekeeper we are not told he knows which box contains the grail - if he does not know and luckily removed 98 boxes that he discovers are empty, or does not open, or even know, but still waiting in suspense like us; and only the narrator knows, then the odds are 50/50
it's a cute problem, but the delivery is flawed so that for each person in the reading, assumptions have to be made - seeing the problem or knowing it in a cultural setting helps one make the right assumptions I suppose as well, but again, that's one of the reasons the problem delivery is flawed..
another way to fix it is for the bridgekeeper to say "I took away 98 of the boxes I knew to be empty"
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Maybe it's a cultural problem in understanding the text but, for me, it's clear that the host knows from the beginning in what box is the SG. You picked it right or not from the begining, he is going to confront you with only two boxes in the next stage. It's very similar to the Three-Doors Monty's problem. If one understand the process there, it's not so difficult to infer that here goes similar.
indeed, there was a tv show, i forgot what's the name, but basically, its like this: you have to choose 1 of 3 doors, in which some of them contain prize, and the others empty. When you pick one of them, the host will take the wrong one, and leaving you with 2 doors, and you can either stick, or change. Suprisingly, you can boost your chance by changing the doors to the other one. Try yourself!
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Yeah! it was called the Monty Hall show, the doors had either a sports car or goats behind them :) its less intuitive with 3 choices too
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The show was actually called "Let's Make A Deal", hosted by Monty Hall.
The answer should be 50. If you have two boxes and one has it in it then it's 50/50. If you reworded the question to ask what is the total probability the grail is in 42 out of all the boxes then it would be 99 according to Bayes' THM.
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You're using faulty logic. Just because there is a situation with two outcomes doesn't necessarily mean both are equally likely to happen. Not every two outcome situation is 50-50. That's the whole trick of the problem and why it causes so much debate. It's not intuitive, but the answer is NOT 50% and this has been explained thoroughly here, on this thread, and in multiple other places. The only other option you have if you still can't see why it's 99% is to find an online program that simulates this game (though it would probably be with 3 choices rather than 100) and watch in amazement as the number of times you don't win tends closer and closer to 66% rather than 50% if you consistently stick with your first choice.
the problem with this equation is in reality its a matter of perspective. if you had stated at the beginning of this equation that the grail itself cannot be removed, your argument would have a leg to stand on. this problem is highly debatable considering that lack of information. it teeter totters between giving the advantage to the cup that hadn't been removed with the others, or "given that those who think will ask themselves well can the grail actually taken out and if so why isn't that stated so hmmm really I have no guarantee so ill go safe and say 50 50 because in reality you cant really know, or does the grail have to stay, and if so chances are it has to be 42." tough equation to sell as intellect. more perception. I struggled with both 99 and 42. they both could be right given the information
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"Then the Bridgekeeper takes away 98 of the boxes which are empty."
Conditional probability would only occur if only if ine event is depent probably on the other so if its declared that the 98 boxes are empty then how can we assume the result of probablity on box 42. For the rest 98 boxes.. So the answer should be 50%.
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You were going down the correct reasoning path, but you ended up with an incorrect solution. It is a dependent set of probabilities. The probability that you selected the correct box originally was 1 in 100. The bridgekeeper removes 98 boxes that are empty. Since the box you chose had 1 in 100 chance of being the correct box, and each of the 98 boxes removes has 0 chance of being the correct box, the only thing left is that box 42 has 99 in 100 chance of being the correct box. 1 in 100 for the one you chose plus 99 in 100 for box 42 accounts for all of the possibilities. (remember all of the removed boxes now have zero chance and we must account for all 100 possibilities.
I'm glad somebody made a Monty Python problem. Thanks!
Don't see how a guy can have a theory on this, when explained step by step like you have, it's basically common sense
99% would only be correct if the choice for boxes were exclusive of yours and chosen randomly hence forwards 50% is actually correct due to the current situation.
The clue doesn't tell you that the thrown away boxes didn't include the Holy Grail though!
Now I get it. I didnt concentrate that 3 was first chosen. Its a tough one.
However, you are assuming that probability is based on previous events. Doesnt work like that. As such, the probability of it eing in box three is 50% and the probability of it being in box 42 is 50%
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Stuart think of it this way. When you selected your box what were the chances you were correct? They were 1 in 100. The fact that you now know the identity of 98 of the other boxes does not change the probability of your original selection - it was 1 in 100. All 100 possibilities still need to be accounted for. Those 98 have now added information and each of them has 0 chance in 100 to be the correct box. The box you chose had (and still has) a 1 in 100 chance. Box 42 now MUST have a 99 in 100 chance of being the correct box because we must account for all 100 chances. You can actually try this with fewer items. Perhaps 5 or 6 and actually do the exercise and record the results. You will see that the probabilities work this way.
If you have 100 boxes take 1 (which is your guess) leaves you with 99% of the boxes left. take away a further 98% leaves you with 1% to chose from. The 98% was taken, your 1% was taken add them together and you get 99%.
I got the right answer since I saw it intutively, but I don't know how to apply the Bayes' formula to get the result in a scientific way and not only intuitive.
Does anyone dare?
what is holy grail
There are 2 possibilities from your first choice:
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The holy grail is in box 3: P = 1 0 0 1 ;
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The holy grail is in one of the ninety-nine other boxes: P = 1 0 0 9 9 .
When the 98 empty boxes are removed, the holy grail stays still. If it was in box 3 (prob. 1 0 0 1 ), it stays in box 3; if it wasn't in box 3 (prob. 1 0 0 9 9 ), it still isn't in box 3. The only difference now is that we know that if the holy grail isn't in box 3, it must be in box 42: therefore, after the 98 empty boxes are removed,
P ( holy grail is in box 42 )
= P ( holy grail is not in box 3 )
= 1 0 0 9 9 ,
which makes for pretty good odds at immortality.
+1 Nice explanation! Thanks
It has to be said that it only applies if the host will always eliminate empty boxes, no matter if you have initially chosen right or wrong, to avoid the hypothesis that he will only do so if you have the right box.
Fully agree, +1
So much this
For this problem to yield 99% as the solution, you have to add in the statement that the Bridgekeeper ALWAYS opens 98 doors and then offers the switch. Suppose you watch the Bridgekeeper offer the 100-box choice to a succession of, say, 50, people without opening 98 doors and offering them a switch. You step forward and make your choice. If the Bridgekeeper now opens 98 doors and offer you a switch, it could well be that he's doing so only because you happen to have chosen the correct door and he wants to lure you to your death.
From what I see the problem stems from the difference between the probability that the grail is in your box, and that it isn't in your box. That's not the same as the probability that it is in another specific box. Let's step by step remove boxes. Out of 100 boxes the probability that it's in yours is 1/100. The probability that it isn't in yours is 99/100 but the probability that it is in any other specific box remains constant with the probability that it is in yours 1/100. Remove 1 box. The probability that it is in yours becomes 1/99 the probability that it isn't in yours becomes 98/99. The probability that it is in any other specific box remains constant with the probability that it is in yours. If you keep reducing you reach that the probability of it being in your box is equal to the probability that it isn't in your box 1/2. However, the issue now is that the probability that you guessed correctly the first time is 1/100, and the probability that you guessing correctly now is 1/2. So the probability of you guessing correctly first and still being correct now is 2 out of 102, 2 chances to guess correctly out of 102 choices. The probability of you guessing incorrectly is 100 / 102. 100 incorrect guesses out of 102 choices. Either box at the end has a 50 percent chance to be correct. But the question is really asking did you guess correct in the first place and did you guess correct in the second place. Mathematically it's more likely that you guessed wrong in the first place, so it is better to change your answer at the end. My math shows a 98 percent chance to be wrong and a 2 percent chance to be right.
Guys It's just as Monty Hall When 98 empty boxes were opened, it's like the opened door with a goat behind it in the tv show Try to see it that way and you'll realize the answer could only be 99%.
Think of it this way: the guy removing the boxes told you that no matter what, when he goes to removing the 98 empty boxes, he will only remove them for the stack of the 99 boxes that you did not choose. Now lets assume that when you picked a box, it was the wrong box (P = 99/100). In this situation, the guy removing the boxes is FORCED to remove the 98 SPECIFIC boxes that do not have the holy grail and leave the one box that has the holy grail there. Since this situation will happen 99/100 times (it will happen every time you choose the wrong box, which that probability is 99/100), the chance that the one box remaining from the pile of 99 has the holy grail is 99%. (This is because the guy removing boxes knows which box has the holy grail and he will not remove the box you chose. Without this knowledge, it is a 50-50.)
The probability you get the holy grail is 1% before removing 98 of the boxes. After removing the boxes you have 99% of winning if you switch and 1% of losing.
Chances that the wrong box was chosen is 99% Chances that the holy grail was chosen is 1% Chances that the holy grail is anywhere else is (100-1)% When its revealed that "anywhere else" means box 42, chances that box 42 holds the holy grail is: (100-1)% = 99%
When you originally choose box 3 you have a 1 in 100 chance of getting the grail. For the other 99 boxes there is a 99 in a 100 chance that the grail is in there somewhere. When 98 empty boxes get taken away, according to Bayes' theorem, the 99/100 probability concentrates on box 42.