Quiz show condumdrum

[firecrest]

This little quiz question sometimes has a nack of causing big rows. Its too long to explain round a camp fire so Ill post it here....

Ok here is the question

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a Prize; behind the others, goats. You pick a door, say No. 1, and before it is opened , the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then asks you if you want to stick with the door you chose, or switch to the other door?" Is it to your advantage to switch your choice?

ps - you dont want the goat!

 

[g4ghb]

mathematically - yes!

 

[Tripitaka]

Graham - how so? The odds have gone up for picking the right door from 33.333% to 50%. Which ever remaining door you choose has an equal chance of being the right one.

So no, I don't believe it is either in, or against, your interests to switch your selection.

Happy to be wrong and gain knowledge.

 

[g4ghb]

ok - looking at it from the other side (of the door?)

Initially you had a 66.6666% chance of loosing - you now only have 50%


practically you are no better off changing

mathematically, using the laws of chance, you are better off changing

 

[g4ghb]

and to be honest - I'm not convinced on this myself! - i'm only doing what i'm told by swmbo (the mathematician)

 

[sam_acw]

Maths says you are better off changing your guess.
First choice you have 33.3% chance of being correct. Removing one outcome doesn't change the probability until a new choice is made.
Second choice is 50% chance.
If you don't make the second choice your odds remain at 33%
In reality you've either got it right or not and you'd feel a right muppet if you changed and picked the wrong one.

 

[Doc]

You chose a door first. The host then chooses one of the two remaining doors. But he knows what is where, and he sure as hell isn't going to open the prize door for you. So I would definitely switch my choice, on the grounds that door number 2 is one the host chose not to open.

 

[hedgepig]

The odds of you picking the correct option from 2 is indeed higher than the odds of you picking the correct option from three ...

... however, there is no "memory effect" with probabilities. There is no advantage to be gained from changing your choice when one incorrect option is removed. Nor is there any advantage to stay with your original choice. Its a simple 50/50.

This is analogous to flipping a coin and getting heads twenty times in a row. What is the probability that the next time you flip the coin is will land on heads? The answer is 50%, the same as always. The coin is just a coin, it isn't affected by what has happened to it before, it doesn't know that it landed heads twenty times in a row. Each individual time it is flipped the chance of it landing heads up is 50%.

 

[Doc]

Actually, the more I think about this, the more I would like to see it explained. I don't buy sam's explanation, but I'm not completely sure of mine either....

 

[Doc]

Found the answer
http://books.google.co.uk/books?id=...&hl=en&sa=X&oi=book_result&resnum=1&ct=result

 

[sapper1]

I have the explanation in one of my books.
It's at the back of "the curious incident of the dog in the night"
Some female mathematician proved it,and she proved that you must change you choice or you''l be guaranteed to lose.
I'll dig the book out and post the theory.

 

[decorum]

Mathematical probablility says you change.
The original odds were that you had a 66% chance of choosing a goat. Seeing behind door 3 doesn't actually lower the odds - they've actually lengthened .
OK that doesn't *seem' to make logical sense does it!
Think about it though ... Your original choice was based on choosing 1 door from a choice of 3. Originally you had a 1 in 3 (33%) chance of the prize and the original odds haven't actually changed but the amount of doors has.
The odds that your original choice was correct is still 33% and not 50% as logic would seem to dictate - so yes probability says changing doors will now give you a 66% chance of winning the prize.

 

[Tripitaka]

But me being the arrogant bottom that I am, doubt the words in that link. It all becomes flawed if you chose the right door in the first place. If you did, then the chance of the prize being behind the second door does not become 2/3 - how can it inherit the odds?

The 14 million doors is just misleading because at it's root, this always becomes a choice from 3.

Still not convinced. But then I'm a statistician, not a mathematician so its practicality rather than theory that matters to me.

 

[hedgepig]

Excellent explanation on Wikipedia - its called the Monty Hall problem:

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

At least I'm in good company

 

[decorum]

... It all becomes flawed if you chose the right door in the first place. If you did, then the chance of the prize being behind the second door does not become 2/3 - how can it inherit the odds?

Although the odds of the original choice being correct lowers it doesn't mean that that it was incorrect - it just means that it has less likelyhood of having been correct. It really sucks if you change and find you were right in the first place though.

 

[firecrest]

Oi youre not supposed to post the answer!!

Yes you increase youre chances by switching. Dont worry if you got it wrong or dont understand, the majority of mathematics professors in America did to.

 

[firecrest]

Maths says you are better off changing your guess.
First choice you have 33.3% chance of being correct. Removing one outcome doesn't change the probability until a new choice is made.
Second choice is 50% chance.
If you don't make the second choice your odds remain at 33%
In reality you've either got it right or not and you'd feel a right muppet if you changed and picked the wrong one.

It doesnt quite work like this though the answer is correct. initially it is 33% but if you switch, it is 75% not 50. The odds are always in your favour to switch.

 

[firecrest]

But me being the arrogant bottom that I am, doubt the words in that link. It all becomes flawed if you chose the right door in the first place. If you did, then the chance of the prize being behind the second door does not become 2/3 - how can it inherit the odds?

The 14 million doors is just misleading because at it's root, this always becomes a choice from 3.

Still not convinced. But then I'm a statistician, not a mathematician so its practicality rather than theory that matters to me.

No no, this is statistics!! Youre forgetting the interplay with the host. His answer affects yours. Let me find summat...

 

[scanker]

I think it helps if you think of 100 doors rather than 3. You pick your 100-1 door. Someone who knows what's behind the rest then opens 98. I'd switch.

 

[Tripitaka]

Firecrest - It isn't statistics as there is no evidence or data. This is maths, as it's pure theory. If someone has data sets that they can send me with the supporting detail then I can look at it statistically. The problem with theory is that I could prove that I was 100% right or 100% wrong.

I'm going to create some data today via a game with the kids - it won't be a massive set but it will be data.

Me? Pedantic? LOL!

As a Lean Six Sigma consultant, I have a motto: "In God I trust - everyone else fetches data!"