Brain teaser

[lay-z]

Not really because:

First Step - you pick a door.
You have a 2/3 chance of picking a donkey
You have a 1/3 chance of picking the car

Step 2 - The host knows which doors contains the donkeys, so, they open the door with the donkey.

Statistically speaking, you are twice more likely to pick a door with the donkey.
If you picked the donkey, the host is forced to open the door with the other donkey.
If you swap, you have inversed your odds from being 1/3 correct to 2/3 correct.
Here's another visualisation. Below are the possible scenarios. Let's run with the assumption you always pick the first option as your door.

Car, Donkey, Donkey
Donkey, Car, Donkey
Donkey, Donkey, Car

So, if you always picked the first option, were revealed where one of the donkeys were, then swapped, you would have twice as much chance of winning the car.

David - I can see where you are coming from with this now (which is where I can now disagree ). From my understandting, you are adressing this hypothetical with a linear approach as opposed to factoring in all the possible outcomes. Is this so?

[James T]

That last point from Alan shows it most clearly I think.

If you switch.
You have a 1/3 chance of picking a car initially - switching and losing.
You have a 2/3 chance of picking a donkey initially - switching and winning.

If you stick.
You have a 1/3 chance of picking a car and winning.
You have a 2/3 chance of picking a donkey and losing.

ie if you are wrong to begin with (more likely) you must win by switching. Whereas if you are right to begin with (less likely) well, hard-lines.

[Xebadir]

But heres the crux of the paradox. 3 people on this forum take this challenge, all follow your advice to switch. Not every time will 2 out of the 3 win the car. You actually need an incredibly large number of samples to trend towards such a result. Take flipping a coin. Take one out now, flip it ten times, do you get 5 heads, 5 tails?

Heres another example: Weather forecasting. Statiscally the 4th day forecast has a probability of 55% of being correct. But on a day to day scale it actually only has a 50% probability, assuming the outcomes are correct or incorrect. Why? You neglect a quantity known as forecast skill of a model. Which effectively acts as a multiplier to your probability. This is more amusing when you consider the 1 day forecast has a 95% probability of being correct, but the actual potential outcomes are 50/50 before one includes model skill :P. Why do weather models get it wrong so often, because eventually the probability you get it wrong happens.

In the above case what is neglected is that no matter what someone loses :P.

Probability in a complex sense is the potential outcomes as the sample size trends to infinity and converges to the result :P.

[Speedway]

Once 1 door has been revealed then it is no longer in the equation so the odds become 50/50, it doesn’t make any difference whether you change your selection or not.
Keith.

[Speedway]

Imagine a game show.
There are 3 doors, one with a major prize and the other two with a donkey behind.
You are asked to pick a door.
You pick one.
The host then opens another door - always revealing a donkey.
The host then asks you - do you want to change your pick?
Do you?

This line changes the original odds whether you change or not.

[ving]

lay-z... the door the host picked and whether or not he knows whats where is completely irrelevant. all that matters is that there are 3 doors, one with a prize 2 with donkeys. if the door the host opened (which wasnt the one you picked) was the one with the prize then the outcome is different... you lost. thats the only other outcome AFAIC up to the stage given in the original post.

[Lance B]

The flipping coin scenario given by xebadir is a good example. If you flipped a coin 10 times and received a head for all those 10 times, what is the likely hood of you getting a head on the 11th flip? Answer = 50/50 as every time you flip a coin it is always a 50/50 chance. However, the odds for getting 11 head in a row are very much higher!

[Steve Axford]

lay-z... the door the host picked and whether or not he knows whats where is completely irrelevant. all that matters is that there are 3 doors, one with a prize 2 with donkeys. if the door the host opened (which wasnt the one you picked) was the one with the prize then the outcome is different... you lost. thats the only other outcome AFAIC up to the stage given in the original post.

You have a choice of two doors. Your original pick has the information that it is a 1/3 chance. The other door has changed information because the host picked one of the donkeys (deliberately) and added new information to that and the other remaining door(but not your door because he would never pick yours. Thus the two doors are no longer random and it does matter which one you choose.

[ving]

The other door has changed information because the host picked one of the donkeys (deliberately) and added new information to that and the other remaining door(but not your door because he would never pick yours. Thus the two doors are no longer random and it does matter which one you choose.

how is the new information passed onto the other door? the fact still remains that it is either a donkey or the prize it doesnt become 2 donkeys or a donkey and the prize....
to quote keith:

Once 1 door has been revealed then it is no longer in the equation so the odds become 50/50, it doesn’t make any difference whether you change your selection or not.
Keith.

[Steve Axford]

The host chooses between the 2 remaining doors, thus the information is imparted to both of those doors, but not to your chosen door (since he can never choose that one). You now have different information for your door and the other remaining door.

[davidd]

Yes, why does the door you didn't pick inherit the other door's chances? I still think both doors will equally inherit the chance, so it becomes 50/50.

You have a choice of 2 doors after the one door is revealed, so the odds are 50/50. You can stick with your original choice (50% chance) or change to the other one, (50% chance).

[Steve Axford]

Because the host chooses only from the two remaining doors and not your pick. Either one or both have a donkey behind. It is 1/3 that both have a donkey and 2/3 that only 1 has a donkey. Therefore it is 2/3 that the remaining door is not a donkey (has the prize) and 1/3 that it is (has not the prize).
He has no influence on the door that you chose, so it is 1/3 as it always was.

This is such a beautiful thing. Our brains get totally bamboozled by it.

[ving]

i am not bamboozled, you are just wrong steve!

[bushbikie]

OK, I'll have a bite....
I wouldn't change my pick. Who's to say that the major prize isn't a donkey?

[Steve Axford]

And you're not bamboozled????? Work out the maths, ving. There are only 3 doors so it can easily be done long hand.

[virgal_tracy]

This is the explanation as portrayed in the movie "21"

Monty Hall Problem

It doesn't quite explain the whys and wherefores but offers some explanation.

Try this as the fulfilling explanation

Monty Hall Explained

I hope this works.

[Ms Monny]

Even after the answer was revealed....people still were talking about it and working it out! WOW, that some powerful brain teaser!!

[ving]

thanks vince. and so you increase your chances of getting the car to 66% if you first picked a goat but if your first pick was the car and you change it is 33% according to the YT vid....
twas an enlightening video makes perfect sense... is completely illogical and IMO still wrong

and now i am going insane!

[Speedway]

No matter which way you look at this the original Odds were 2:1. When the host opened one of the other doors to reveal a donkey your odds were still 2;1 but when he offered you a chance to change your selection the third door was removed from the equation leaving only two doors, and whether you changed or not because you were given the opportunity your odds are now 1:1 or 50/50. If you were not given the chance to change and he was just prolonging the agony the odds of you having picked the correct door are still 2:1
Keith.
PS Some people think too hard finishing up only confusing themselves.

[Steve Axford]

Wrong, wrong, wrong. That's what I spent a day hurting my brain with. It does matter - the doors are now in different groups. We assume that they are all equal, because that's the way they started out. But the host changed all that when he opened a door which he knew was a donkey.
As Ms says - wonderful brain teaser because even when explained, many of us still don't believe it.