Unexpected egg paradox

[Luke]

the theory:

i play a game with you. i present you with two boxes. i say to you that in one of the boxes is an unexpected egg (you will see what this means in a second), the only rule of the game is that you must look at the boxes in chronological order. so now you think, well it cant b in box two, because if you opened box one and nothing was in it then you would know for sure that the egg was in box 2, thus making the egg expected, hence the egg must be put in box 1 in order to be unexpected, thus making it obvious that the egg has to be in box 1, therefore th egg loses its unexpectedness. we have an impossible situation.

 

now lets open the game up to ten boxes, still you must look at them in chronologica l order. i say i put an unexpected egg in one of the boxes. you think, hmmm well it cant be in box 10 because if i get all the way up to box ten without seeing an egg then the egg has to be in box ten, thus making it unexpected. so i cant put it in box ten. so now you get all the way up to box 8 without seeing an egg, and because the egg cant be in box 10, then it has to be in box nine, thus again making it unexpected. im sure you can see the pattern now, so now we repeat the logic all the way back to box one, thus making it logically impossible for me to put an unexpected egg in one of ten boxes.

 

My question: if this were to be done in real life, you would go through the boxes, and booom its unexpectedly in box 6, so wheres the logic going wrong?

[Moontanman]

How can it be unexpected if you know it's in one of the boxes?

[BrianG]

Is the egg hard boiled, soft or raw?

[lemit]

Why again would the egg be expected if it were in box 1 of 10?

 

Thanks.

 

--lemit

[Moontanman]

The egg is poached, if the egg is assumed to be in one of the boxes then is it expected. If I opened a random box with no knowledge of the possibility of the egg only then would it be unexpected.

[lemit]

So, what leads me to expect the egg to be in box 3?

 

--lemit

[modest]

I've known this as the "hanging paradox"

 

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on a Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on a Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.

Unexpected hanging paradox - Wikipedia, the free encyclopedia

I've just read this chapter:

The Colossal Book of Mathematics -- Chapter 43

which gives an explanation and multiple examples. It's really quite a fascinating paradox.

 

At the heart of the paradox seems to be a self-referencing self-contradicting statement such as:

The next sentence is true. The previous sentence is false

If a person were to assume that the above quote is truthful and accurate then a paradox ensues. Likewise, if I assume that the following is truthful and accurate,

i present you with two boxes. i say to you that in one of the boxes is an unexpected egg (you will see what this means in a second), the only rule of the game is that you must look at the boxes in chronological order.

then a paradox ensues. By assuming that the prediction of an unexpected egg is true then I can neither logically consider the second box nor the first box to contain an unexpected egg.

 

Whatever probability we give to the egg being in the first or the second box is changed by the act of assigning the probability. If I assign 100% probability that the egg is in the second box (which seems reasonable if the first box has been opened already and no egg was found) then there should be 0% probability that the egg is in the second box—because 100% probability makes it expected and we are assuming that it is not expected.

 

The probability references itself and contradicts itself. Very interesting paradox.

 

~modest

[lemit]

Thanks. I get it now. It works in part, it would seem to me, because the prison population are generally not the best and brightest.

 

I hope his possibly faulty logic allowed the prisoner some peace in his final days. I know I'd be too terrified to engage in mental exercises.

 

I'm curious about something. For his last meal, did the prisoner request boxed eggs? No, wait. By keeping the day of the execution secret, the judge denied the prisoner his last meal, the chance to say his last goodbyes to his family and friends, and most of all, his chance for last-minute appeals. The prisoner's lawyers--on Sunday--probably got a stay of execution and eventually a mistrial. So the question then becomes, was the judge surprised?

 

--lemit

[Qfwfq]

In the terms of the original unexpected hanging paradox, it boils down to the prisoner's mistake of confusing the conclusion "therefore the hanging can't be unexpected" with the erroneous one "therefore I won't be hanged". Indeed, the prisoner is hanged, but the statement that the day on which it happens will be certainly unknown until it does happen is (although not so obviously) contradicted by the fact that it will occur within a fixed, finite number of days.

 

In order to rule out Thursday on the grounds that it can't be Friday, the word certainly is essential and likewise for the other days until the first. The executioners do not care about the prisoner's nitpicking precision and so his argument does not save him. Logically, the contradiction is there, but the sentence must be upheld all the same so it's precise and strict logic that falls into neglect.