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Skyhook

The Monty Hall Dilemma

(For those of you who remember the plane on the conveyor).

You are on a game show.

You are presented with three doors. Behind one is a car, that you will win if you pick the right door. Behind the other two are goats.

You pick a door.

The gameshow host will now pick a door to open; the only rule being he can't choose your door or the one with the car behind it. So, he opens a door and reveals a goat.

So two doors are left. The door you first picked, and the other. One definitely has a car behind it. The host offers you the choice to change your door.

What do you do? Stick with your first choice or change?  Statistically, which door now gives you the best chance of winning the car?
Frank Bullitt

Evens, but I'm probably going to find out in 15 pages why that can't be right.
PhilD

May the odds be ever in your favour.

I know the answer but won't spoil it. It's a bit of a mind fuck.
Skyhook

When you picked your door, you had a 1 in 3 chance of the right answer.

So your door has odds of 1/3.

The host removes one option.

The remaining door therefore  has 1/2 odds of being right.

Which one one is the best choice now?

(Yes I know, you like me are probably thinking it's 50/50, but bear with it, cos we are both wrong.)
Michael

I think I read something on this, your better swapping doors aren't you?
Big Blue

The odds depend on whether you're looking at the second choice as a single choice (50:50) or as part of the overall game where you know the host could not select the car or your door, which may have been one and the same. I'm no good at that sort of odds definition but that's the scenario where it's not 50:50.

I'm guessing if you keep your door it remains a 33:66 choice as that's what it was in the first place whereas if you swap doors it's back to 50:50 so theoretically you're better off swapping doors but of course that's theoretical bollocks as the choice is back to 50:50 in the second round as it's simply a choice of two doors either of which could have the car behind it.

The alternative scenario is that you're an Arab Nomad and want the goat.
TreVoR

James May explained it with fizzy lager cans on Man Lab. You are better off changing.
Bryan M

It is something to do with the odds of 1in3 mean you are less likely to have chosen the winner and therefore the odds are the host is choosing between the winner and looser, so by swapping you are increasing your chances
PhilD

Big Blue wrote:
The odds depend on whether you're looking at the second choice as a single choice (50:50) or as part of the overall game where you know the host could not select the car or your door, which may have been one and the same. I'm no good at that sort of odds definition but that's the scenario where it's not 50:50.

I'm guessing if you keep your door it remains a 33:66 choice as that's what it was in the first place whereas if you swap doors it's back to 50:50 so theoretically you're better off swapping doors but of course that's theoretical bollocks as the choice is back to 50:50 in the second round as it's simply a choice of two doors either of which could have the car behind it.

The alternative scenario is that you're an Arab Nomad and want the goat.


Or the car doesn't have a full size spare wheel so you want the goat.

BB it's not theoretical bollocks, it's what will happen - the "second round" is not 50:50. A new first round with two doors (where you knew nothing about the other game) would be.
Humphrey The Pug

Shouldn't this be in the "Random" thread?!
Stuntman

You definitely switch in my opinion.

To start with you have a straight 1 in 3 chance.

The host now has to open a door with a goat behind it, so it means that your door either has the car behind it, or doesn't.  

The host has to choose between two doors only.  There will either be a goat and a car behind the two doors (2/3 of the time), or two goats (1/3 of the time)

So therefore, 2/3 of the time, the door he doesn't open will have the car behind it.  You are now being offered a 50/50 switch, but 2/3 of the time (in the long run) it will be correct to take up the offer.  Therefore the best 'expected value' decision is to switch.

I think that makes sense!  
Chris M Wanted a V-10

Big Blue wrote:

The alternative scenario is that you're an Arab Nomad and want the goat.

Shouldn't that be an Arial Nomad, in which case it's the car anyway...... ?
Roadsterstu

Even if the odds were heavily in my favour, I'd probably still get the goat.
Twelfth Monkey

If the host can't pick one with a car behind it -

If it is two goats, there is a fifity/fifty chance that the door he doesn't choose has a car behind it.

If it is a car/goat choice he has, he has no choice.  The other door has a 100% chance of having a car behind it.

Two goats is originally more likely (and is still more likely even after you've chosen), but averaging these still gives a probability greater than 50% that the door neither chose hides the car.

Is that right?


(I still think that this involves nesting probabilities though, so I think it's 50/50 in reality - is anyone daft enough to try this out?)
Bob Sacamano

Being the simple person I am I would look at it as; when you start you have a 1 in 3 chance of being right and if you don't do anything else you still have a 1 in 3 chance. You're then offered a chance to increase your odds to 1 in 2 so you switch and take it.

I know that's not the right explanation though.
gonnabuildabuggy

Michael wrote:
I think I read something on this, your better swapping doors aren't you?


My thought too. Game theory, as explained by James May on TV a while ago.

My kids have actually done it in school.

Now I'll scroll further down to see if I'm right (this was my first thought and i've only scrolled down to Michaels' comment so far)....

EDIT. I think it's more complicated as he can't open a door with a car behind it or yours. So in fact the original odds were something like 50/50.

Definitely switch though as he had better odds than you before, now it's evens.
PG

Never mind all that. Will the car take off is surely the most important question?
Twelfth Monkey

Hmm.

1/3 of the time, you have a car and will have a 100% chance of losing it.

2/3rds of the time, you have a goat and will have a 100% chance of getting the car.

Swapping therefore gives you a 2/3rds chance of winning the car.


Although I am talking myself out of this as I type...
PhilD

Bob Sacamano wrote:
Being the simple person I am I would look at it as; when you start you have a 1 in 3 chance of being right and if you don't do anything else you still have a 1 in 3 chance. You're then offered a chance to increase your odds to 1 in 2 so you switch and take it.

I know that's not the right explanation though.


Its not far off. If you think of a million doors it helps. At the start you had 1 in a million change of being right. When it's down to 2 doors your choice is still a 1 in a million shot. The other door started off as a one in a million but by opening the other 999,998 doors the host reduces it to 50:50.

So would you stick with your 1 in a million shot or swap to a 50:50 one?
PG

I just read a big write up on this online and now my head hurts. I think it all comes down to how you measure and interpret probability.  

When you pick a door (call it door 1), you have a 1/3 chance of being right. The other pair of doors (2&3) have together a 2/3 chance of being right.
But when the host has to open an empty door he reduces the pair of doors with 2/3 probability to one door - that still has 2/3 probability.  

Solution 1 - so you ought to switch as you are swapping from 1/3 (door 1) to 2/3 (the other door from doors 2&3) probability

Solution 2 - yes as a point solution that is correct, but there is still a 1/3 chance that the car is behind door 1. So the mathematical solution of choosing or switching the door is (1/3+2/3) divided by 2 = 1/2.

Take you pick......
woof woof

Why is everyone assuming the contestant want to win the car?

There may be many reasons why the goat could be the best prize.

I can think of one...
Roadsterstu

woof woof wrote:
Why is everyone assuming the contestant want to win the car?

There may be many reasons why the goat could be the best prize.

I can think of one...


I daren't ask.
gooner

One might want to stare at it
Stuntman

I think he's kidding...
PhilD

Stuntman wrote:
I think he's kidding...


Of course he is, missing out on the car would really get one's goat.
Roadsterstu

I could have been gruff but thought it best to be polite.
Michael

Now you're milking it.
Bryan M

Don't nanny him, he's not a kid.
Twelfth Monkey

Cheese-us, another pun run...

I wouldn't mind be the quality of puns is so up and down - it positively ungulates between great and awful.

https://en.wikipedia.org/wiki/Ungulate
Michael

Check out the big word from Billy no mates.
Twelfth Monkey

You know when you're looking a gift goat in the mouth!

(I only know it because it was in a Far Side many, many years ago.)
Michael

So you thought you'd hoof it in?
PhilD

Michael wrote:
Check out the big word from Billy no mates.


Why are you having a go at 12th?
Michael

PhilD wrote:
Michael wrote:
Check out the big word from Billy no mates.


Why are you having a go at 12th?


Twelfth Monkey

Sniff.

I'm used to a bit of trolling...
Bob Sacamano

Just another thread symptomatic of today's Nanny State.

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