brain teaser

[kityuser]

This interesting brain teaser was suggested at work, several IC design engineers argued over the answer for several days.......

" a game show host as 3 boxs, one has a prize in. The contestant is made to pick a box. The gameshow host then purposefully opens up an empty box leaving 2 (the one the contestant has and the remaining box). Will the contestant have more chance of winning by gambling to swap for the remaining box or sticking with his/her original choice?"


answers on a postcard please......


8)

Steve

 

[Karl]

Well here's my view:

When all 3 boxes are lined up, the contestant has a 1/3 chance of picking the box with the prize in. Meaning that the two left behind have a 2/3 chance of containing the prize.

When one of the boxes is opened, which isn't the contestants, and turns out to be empty, the odds of the remaining box containing a prize are still 2/3.

So the box which is not the contestants has a 2/3 chance of containing the prize, and the contestants box has a 1/3 chance.

So he should swap.

Cheers

Karl

 

[Balbus]

It won't make any difference. Once one box has been opened, and therefore discounted, the chance of the box the contestant chose containing the prize is one in two.

If he changes his mind and swaps, it's still one in two.

 

[RogerS]

Karl is correct. The contestant should always swap. This teaser was aired on Radio 4 the other day and up until that point, I'd always thought that the chances were even.

 

[kityuser]

So the box which is not the contestants has a 2/3 chance of containing the prize, and the contestants box has a 1/3 chance.

not sure how you arrived at this one....


Steve

e

 

[rkchapman]

He should swap.

Call the three boxes A B and C, with the prize in A.

There are 3 possible cases. If he picks A at first, then swaps, the swap was a bad idea. If he picks B or C at first then swaps, the swap was a good idea. So it's a good idea more often than it's a bad idea.

 

[kityuser]

Karl is correct. The contestant should always swap. This teaser was aired on Radio 4 the other day and up until that point, I'd always thought that the chances were even.

Karl is correct, although I`m to be convinced of the reasoning.

I solved it by drawing a quick probability tree and (sadly) solving it mathematically.

The important point to note is the gameshow host ALWAYS opens an empty box. Therefore this sways the odds as it is'nt a random choice.........


oh the arguments at work were funny..... :roll:


Steve

 

[RogerS]

You have to look at it from the point of view of the gameshow host. He knows which is the contestants' box.

I agree that Karl's explanation isn't quite right (I think) but the bottom line is it's better to swap. Or was it to stay? Ummmm.....


EDIT: Yup..I'm right. http://www.mitan.co.uk/library/gameshow_problem_v4.pdf

 

[Karl]

The reasoning:

The odds of the show remain static from the moment the box is chosen - they are not re-calculated each time a box is "eliminated".

So if your box has a 1/3 chance of winning at the start (and by definition the others have a 2/3 chance of winning at the start), those odds remain the same even though 1 of the 2/3 is eliminated.

Hope that makes sense.

Cheers

Karl

 

[kityuser]

The reasoning:

The odds of the show remain static from the moment the box is chosen - they are not re-calculated each time a box is "eliminated".

So if your box has a 1/3 chance of winning at the start (and by definition the others have a 2/3 chance of winning at the start), those odds remain the same even though 1 of the 2/3 is eliminated.

Hope that makes sense.

Cheers

Karl

yup

Steve

 

[Balbus]

Ah I see . . .

Actually I don't really.

But Keith Devlin's explanation is pretty convincing (http://www.maa.org/devlin/devlin_07_03.html) and he's very, very clever.

 

[Anonymous]

Even after reading the article I'm still not convinced. I believe some of the arguments are flawed. I do understand that history makes a difference to probability though. This is shown by the coin flipping question:

If you flip a coin what are the chances of you predicting the result? Now do it 99 times and record what happened. On the hundredth time can you predict the result any better?

OK, maybe I didn't put that very well but I'm sure most of you get the gist.

Dave

 

[Tom K]

Damned mathmaticians :shock:

http://en.wikipedia.org/wiki/Monty_Hall_problem


How about if you transpose the problem to Russian roulette would you trust the math?

 

[TrimTheKing]

Karl's reasoning is spot on.

This was also featured in the recent Kevin Spacey film about card counting (can't remember the name, might have just been 21) and the reasoning behind it was exactly as Karl wrote... :wink:

Having said that, there is no guarantee that the film got it right, but my missus who has a degree in Mathematics and Econometrics (and a generally monumentally huge brain) and some general googling convinced me that it was correct.

That still doesn't make it right though And I still don't understand it fully :lol:

 

[kityuser]

Karl's reasoning is spot on.

This was also featured in the recent Kevin Spacey film about card counting (can't remember the name, might have just been 21) and the reasoning behind it was exactly as Karl wrote... :wink:

Having said that, there is no guarantee that the film got it right, but my missus who has a degree in Mathematics and Econometrics (and a generally monumentally huge brain) and some general googling convinced me that it was correct.

That still doesn't make it right though And I still don't understand it fully :lol:

The best thing to do is get a GCSE mathematics book, look up probability "trees" or "maps" and give it a try.
Not trying to be patronising, its really not that hard when you draw it out and it dawns.

Steve

 

[TrimTheKing]

The best thing to do is get a GCSE mathematics book, look up probability "trees" or "maps" and give it a try.
Not trying to be patronising, its really not that hard when you draw it out and it dawns.

Steve

That's the thing, me and maths have never really seen eye to eye. But when it does 'click' I have one of those eureka moments and beat myself up for ages for not picking it up sooner.

I will investigate...

 

[Russ]

If a plane crashed on the U.S and Canadian border - Where would they bury the survivors?

 

[kityuser]

If a plane crashed on the U.S and Canadian border - Where would they bury the survivors?

you don't bury survivors

Steve

 

[Karl]

The best thing to do is get a GCSE mathematics book, look up probability "trees" or "maps" and give it a try.
Not trying to be patronising, its really not that hard when you draw it out and it dawns.

Steve

That's the thing, me and maths have never really seen eye to eye. But when it does 'click' I have one of those eureka moments and beat myself up for ages for not picking it up sooner.

I will investigate...

I was expaining this quandry to SWMBO this morning, and relating it to Deal Or No Deal, which it is clearly based on. Imagine there is £250k left on the board, and 1p. To play devils advocate, the dealer has chosen not to make an offer - you must choose one of the boxes as your prize.

Her answer was to stick with the box she's got. When I pointed out to her that, in that show, the odds of the top prize being in the box she has chosen are 1 in 22 (4.6% chance), and the odds of it being in the other box are 21 in 22 (95.4%), she said that she would still stick with the box she had.

:shock:

So why?

"Because I could bear not winning it if I never had it, but if I had it and then gave it away, i'd be suicidal"

That's women for you :lol:

Cheers

Karl

 

[Steve Maskery]

I was never convinced of the arguments either until it was put to me this way.

The key is that the Host knows where the prize is. He is offering you a 50/50 deal, when you made your original choice on a 1/3 deal. He's not opening one box, he's opening ALL the boxes except the Prize box and your choice.

Let's take the case where there are not 3 boxes but 1000 boxes.

You choose no. 483. The probability of you being right is 1:1000

Ah! says the Host, 483, eh? Sure you don't want to change to 219? And he opens all the boxes except 483 and 219.

What do you do then?

You should always swap. You won't always win, but you are more likely to do so.

Cheers
Steve