3 Boxes Problem

[Lewi T]

No idea why this problem came into my head, but it did so I was having a little browse.

But anyway, what I'm thinking is that ultimately, you're not making a decision until there's two boxes; one winner, one loser. Is it not irrelevant which box you pick first?

The diagram that you posted Kyle, should there not be 4 choices? Option 1 could be split into two as there's two possibilities for which un-chosen, empty cup to remove. That would make it a 1/2.

I've always leant towards it being a 1/3, but for some reason I can't decide now, which annoys me.

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[Napalmbrain]

No idea why this problem came into my head, but it did so I was having a little browse.

But anyway, what I'm thinking is that ultimately, you're not making a decision until there's two boxes; one winner, one loser. Is it not irrelevant which box you pick first?

The diagram that you posted Kyle, should there not be 4 choices? Option 1 could be split into two as there's two possibilities for which un-chosen, empty cup to remove. That would make it a 1/2.

Gosh, I thought you were doing a maths degree? Yes, option 1 is technically split into two, but it's irrelevant which one the host removes because you know it's wrong.

[Lewi T]

Gosh, I thought you were doing a maths degree? Yes, option 1 is technically split into two, but it's irrelevant which one the host removes because you know it's wrong.

Would that not create another possibility though?

[Sovieto]

Probability aside, if this situation ever comes up, it'll be on Wheel of Fortune or some other yawntastic game show or a street-side entertainer. This will be only ONE instance, in which probability doesn't play a factor or matter a shit in my opinion. My advice would be to go with your gut instinct and keep the first one you picked.

[ssbb_lover]

Probability aside, if this situation ever comes up, it'll be on Wheel of Fortune or some other yawntastic game show or a street-side entertainer. This will be only ONE instance, in which probability doesn't play a factor or matter a shit in my opinion. My advice would be to go with your gut instinct and keep the first one you picked.

Pretty sure Wheel of Fortune only deals with words.. and just because it may only happen once in a life time, doesn't mean you shouldn't apply the same logic you apply to everything else. In fact, there me more of a reason to do so being a rare occurrence. Depends on how you look at it I suppose.

I think my original way of figuring this out was flawed, even if it was correct. I'm not sure I completely understand.. do you have a 2/3 chance at the end solely because one of the cups has been shown empty?

[Napalmbrain]

Would that not create another possibility though?

Not really. Basically, there are three choices: you can either pick the "good" box, or one of the "bad" boxes. If you picked the one of 2 "bad" boxes, then switch, you win. But if you picked the 1 "good" box, then switch, you lose. So assuming you switch, 2 possibilites result in a win, and 1 possibility results in a loss.

[Lewi T]

Excellent. I was hoping to be completely proved wrong, because I knew 2/3 was correct but I'd been looking for proof that switching is best, but everywhere I had looked people couldn't completely validate their answer.

[Sterculius]

It's all the same to me, because while the chances of success may theoretically change, the item within the box never changes boxes from start to finish. So while your chances get better or worse it's still all down to chance, no matter how many you're choosing from. Unless you're psychic.

[ROB64]

Bloody hell, never thought of it like that. Interesting...
I guess I should have taken statistics in my fifth year of maths instead of stopping at four years.

[LevesqueIsKing]

Oh, I get it. Because the box he removes can never be the box that you originally chose, so the yet-to-be-chosen box has 2/3 chance of being correct?

That took me forever to comprehend, and part of me still wants to believe that it's simply 50/50, but it's okay because I have yet to take statistics.

EDIT: That was a really interesting explanation, Napalm. That you want to originally pick a bad box so that you can switch to the good.