You are playing a game. A person (game host) shows you 3 boxes. 1 box contains £100, 2 others are empty. You need to choose one of the boxes. If the box you chosen contains the money you win. Suppose you make a guess and point at some box. The game host instead of opening the box you just pointed at opens one of the other boxes and shows you that that box is empty. Then he asks "I'm going to give you a last chance to change your decision". So you can either stand by your original choice and point at the same box or choose the other one. What gives you better chances? (i.e. what are the chances you win if you choose the box you originally selected vs. the other box?)

I'd stay with my original choice. It's 50/50 either way, so it doesn't really matter unless you have x-ray vision.

Haha, morons. edit: Think of it as two sides. There's your side, and the host side. When you choose one, the chance of having the hundred pounds in your side is 33% and his side is 66%. When he opens one of his and shows that it's empty and asks if you want to switch, you say yes, because his side still has a 66% chance of being the good box. Edit: In case the first line seemed too offensive, know that I thought this at first as well.

Wow, I was actually leaning towards Adam's answer. Definitely thought the same as y'all at first though. You guys should know that the answer isn't going to be that simple if Lewi posts an entire thread for it.

Damn, Adam beat me to it. Yes, it is statistically better to change your answer (I've seen this problem before, with doors instead of boxes).

I find this difficult to understand what you guys are saying? 3 boxes, 33% chance of the £100 being in any of them. But if he shows you that 1 of the boxes are empty, then it's a 50% chance that your box is an empty one compared to the other 1.. So switching it would still make a 50% chance surely. Unless you do all that 1/3 times a 1/2 statistics tree thing, but even that would come out the same?

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.

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.

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 **** 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?

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.

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.

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.

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. :lol:

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.