For those more maths-oriented than I

[kangeiko]

I got the following thought out of the book I'm currently reading, Sixty Days and Counting -

You start off with three boxes, one of which contains a ten pound note. The boxes are all closed. You choose one box, and get to hold on to it (still closed). Of the two boxes left, the dealer opens an empty box. You can then choose to swap boxes or not. What should you do?

Now, to me, this was a straight 50-50 - the note is either your box, or the other box. However, there's another answer given -

Each box has a 1/3 chance of containing the tenner. Therefore, when you choose your box, it has a 1/3 chance, and the other two boxes have a combined likelihood of 2/3 chance. So far so good. However, once you open the empty box, the probability doesn't drop... so you have 2/3 probability concentrated in the other box, versus the 1/3 in your own box. You should always therefore switch.

Now... this makes sense. However, I think there's a fallacy in there somewhere. Does opening up one of the boxes constitute an outside event that means you are at a new event nodule and have to recalc probabilities again? Or is the above argument correct, and you are still in the same event?

Comments

[lilka.livejournal.com]

The answer they give is correct. There's a similar puzzle in The Curious Incident of the Dog in the Nighttime, but they don't open either of the remaining boxes (or doors, in that example), and you should still always switch because it's mathematically provable that you're more likely to win by switching than by sticking with your original box. By opening one of the boxes, and therefore eliminating one of the incorrect choices, they actually increase your chances of winning by switching. I just drew out a really long diagram to check this and it's definitely right.

[kangeiko]

Thank you! Worryingly, I passed the exam that had these sorts of questions... despite getting them all wrong. *hangs head* Which says something about the state of education nowadays...

*lightbulb* someone should tell the people at 'Deal or No Deal'!

[darlas-mom.livejournal.com]

I suck at math problems of any kind, but this one amuses me, because here in the states--I wish I was making this up--we have a game show based on this math problem.

[kangeiko]

Same here! *g* It's not 'Deal or No Deal', is it? I hate that show with a fiery fiery passion... possibly because it's hosted by Noel Edmonds...

[darlas-mom.livejournal.com]

LOL. I'm not fond of the show, either, mostly because game shows in general bore me and because my entire family (on my mother's side, anyway) is obsessed with it, so there's no escape.

I don't know who Noel Edmonds is. :-/ Over here, it's hosted by Howie Mandell.

ETA: P.S - are you around in any IMing sort of fashion?

[kangeiko]

I'm on gmail at the moment... have tried pinging you... *crosses fingers*

I don't know who Noel Edmonds is. :-/

Spawn. Of. Satan.

[http://users.livejournal.com/__marcelo/]

One way to look at this is without making the actual probability calculations that they are offering you two different ways to pick your box:

1. The "normal" way, in which you select a random box.

2. The "special" way, in which you select a random box, but with the added guarantee (due to opening up one of the empty boxes before the switch) that you won't be jumping to at least one of the two empty ones.

Clearly, the "special" way is better, because you can either land in the right box or the unopened empty box, while the "normal" way has twice the number of empty boxes for to you pick.

Dunno if this made much sense. :)

[athena25.livejournal.com]

Clearly the best way is to bash the dealer over the head with their empty box and make off with the two other boxes. After rifling through the pockets of the dealer for spare change and credit cards.

Noel Edmonds is hardly the spawn of Satan. He's a bit, damp. But not actually evil. Not like David Cameron.

[http://users.livejournal.com/__marcelo/]

I bow to the superiority of your tactical skills. *bows*

[athena25.livejournal.com]

The trick is to remember that it's only a maths problem if you let it be a maths problem.

Sometimes it's a maths opportunity...

[http://users.livejournal.com/__marcelo/]

:D I'll keep that in mind.

[kangeiko]

*thinks*

My brain hurts.

*whimper*

Going to have some tea, methinks...

[jekesta]

Both mathematically and experimentally you'll always be better off changing your mind because the odds just got better. It infuriates me quite a lot because it shouldn't be true, I see that it is true, but it's SO WRONG and hm. Yes.

[kangeiko]

*blink* Brain!death, I feel. Mmmm, tea&cake...