The Monty Hall Dilemma........goats And Caddys!

[ToraToraTora]

http://www.bbc.co.uk/news/magazine-24045598
I have come across this before and like most initially I always thought that switching or not made no difference but once the penny dropped I saw it. I have had many a difficult time trying to explain why you should switch to people and my explanations have been somewhat clumsy but this explanation is as neat as I have come across:
"In the game you will either stick or switch. If you stick with your first choice, you will end up with the Caddy if and only if you initially picked the door concealing the car. If you switch, you will win that beautiful automobile if and only if you initially picked one of the two doors with goats behind them."
So do you get it? Never got it? don't accept it? ......

[factor-fiction]

Yes, I'm very familiar with it. I recall my natural instinct was to say 'it makes no difference' until I thought through the maths.
I have some similar ones if you are interested n them.
There is a Monty Hall problem simulator site, by the way

[boxtops]

My brain hurts....

[jake-the-peg]

They key point in this problem is that Monty (the game host) knows which door has the car and which doors have the goats.

By opening a door he is supplying extra information.

Monty never opens a door that you did not choose to reveal the car.

If Monty did not know this and could open a door that is in front of the car the strategy fails and you would be as likely to win sticking as switching

[Hymie]

That is a very good explanation as to why if you stick, you have a 1 in 3 chance of winning the Caddy and a 2 in 3 chance of winning the Caddy if you switch.

Look at it this way, assuming you have a choice of three letters A, B or C – picking one of the letters wins the Caddy. Let’s imagine the winning letter is A.
If 30 people pick a letter at random and stick with their choice, on average 10 people will pick A, 10 B and 10C. So on average 10 of the 30 will win the Caddy.

Now instead of sticking, all thirty switch their selection (as per the Monty Hall game), all those that selected A will loose, because they will swap away from a winning selection (choosing B or C). But all those who selected B or C will win, because for those who selected B, Monty will show them C, forcing them to swap to A, and those that selected C will be shown B, forcing them to select A. And therefore, in the group of thirty who swapped, on average they will win 20 Caddys between them – rather than 10.

[jim360]

I've never had a problem with it, but you have to be careful to state the problem carefully. Everything relies on what the host knows about the doors. If he knows where to find a goat, then hes only ever going to open a goat, so in a way the door-opening is a distraction from the fact that you had a 2/3 chance of losing in the first place. If he doesn't know, then he could also end up opening the car door and the odds change again -- from about 67% win by switching to just 50%. And if you don't know where the host knows or not... I've not worked out that case yet, but I think you'd do best to switch anyway, because it's either 50-50 or 2/3, so should end up being somewhere in between!

[Gizmonster]

Jim I don't quite follow your post.
I was with you until:
"If he doesn't know, then he could also end up opening the car door and the odds change again -- from about 67% win by switching to just 50%."
If you're saying that the prize is a car, then surely opening the car door, reduces the chance of you winning down to 0% (assuming there is only one car) ??

[Peter Pedant]

agree Jake

By opening the goat door the show host is adding info
and also making a choice

otherwise half the time he would open the goat door and half - the one with the Roller in it oops.

Therefore it is a 'new game'

[Ellipsis]

It all becomes a lot simpler to understand if, rather than three doors, there are 10,000

You are asked to choose one of 10,000 doors that the prize may lie behind.

The host, knowing where the prize lies, then opens all the other doors bar one - a total of 9,998 doors all with no prize behind.

You are then given the option of sticking with your initial choice, or swapping - what do you do?

Clearly in this case you swap.

And the exact same logic applies no matter how many doors there are, from three upwards.

[jim360]

I might have to redraw the tree gizmo, but I think the gist is that if the host had a chance of opening the car door then it must change the odds. The point is that if he is always going to open a goat door no matter what you choose then that event acts as a kind of distraction. Really you need a pen and paper to see what's going on, draw a probability tree for all events in the two general cases where the host either knows, or does not, what is behind each door.

[Ellipsis]

In the Monty Hall Problem, the host knows which door the prize lies behind and has no chance of opening it.

[mibn2cweus]

In the 'Monty Hall' scenario; following the contestants initial selection (guess), Monty is compelled to eliminate a goat from the other two candidates, consequently doubling the probability that the remaining option is the coveted Caddy . . .

At the onset all three doors are equally = Goat 2/3, Caddy 1/3

Following contestants initial selection:

Guess = Goat 2/3, Caddy 1/3
Other two doors = Goat 2/3, Caddy 1/3 + Goat 2/3, Caddy 1/3 =
Goat 4/3, Caddy 2/3

Monty invariably eliminates one Goat leaving the remaining preferable option =
Goat 1/3, Caddy 2/3!!!

Monty has effectively eliminated one Goat (and one door) from the remaining field of alternative options.

[ToraToraTora]

all good stuff above, interesting point from elipsis, that does help if you stretch it to a lot more doors. Yes monty always opens a goat and knowing that from the outset you must switch to get the best chance of the caddy!

[vascop]

When you choose a door initially, you are twice as likely to have chosen a goat than the Cadillac, so given a choice to swap with a 50-50 chance you should take it.
I like hymie's explanation.

[chakka35]

The woman who is reputed to have the highest IQ in the USA had a magazine column in which she explained why changing your decision doubled your chance of winning.
She received many letters of criticism and abuse. One of them, astonishingly, was from a professor of mathematics who was furious! He accused her of misleading the youth of the country in matters of probability and said she should be ashamed of herself.
I never did find out whether he ever sent a grovelling letter of apology when he discovered what a prize fathead he had been.

[nescio]

another simple way of looking at it is this:
the probability that you initially select the door with the car is 1/3. Therefore, the probability that the car is behind either one of the other two doors is 2/3. This cannot change. The host simply reveals another door which the car is definitely NOT behind. The remaining door still has a 2/3 probability of concealing the car. Ergo, you double your chance of winning by switching to that door.

[Prudie]

I'm also familiar with it and although my head still thinks there is no difference I can follow the maths and can see no flaw in it so it must be correct. A very interesting conundrum.

[LazyGun]

Yeah, I get it. Excellent practical demonstration of changing probabilities, and humans poor ability to properly comprehend odds etc.

Best example I was involved - I think- Marcus Du Sautoy and one of the more popular TV comedians. One always swapped, one always held to their initial choice - over a series of 100 selections, the person switching was significantly better off in terms of prizes. Just cannot think which TV programme it was ;)

[mibn2cweus]

chakka35
The woman who is reputed to have the highest IQ in the USA had a magazine column in which she explained why changing your decision doubled your chance of winning.
She received many letters of criticism and abuse. One of them, astonishingly, was from a professor of mathematics who was furious! He accused her of misleading the youth of the country in matters of probability and said she should be ashamed of herself.
I never did find out whether he ever sent a grovelling letter of apology when he discovered what a prize fathead he had been.
11:51 Thu 19th Sep 2013

One should avoid choosing what is behind door number four, at all costs.

http://corporatecognewdaddyblog.files.wordpress.com/2012/12/2hr1g5c.jpg?w=497

Perhaps the most valuable lesson that can be taken away from all of this is the realisation that someone in a so called "position of authority" is not always right.

http://www.gpposner.com/vos_Savant.html

[Ellipsis]

Good link from mib above, especially as it contains my explanation, but worded better ...

"Suppose there are a million doors, and you pick door No. 1. Then the host, who knows what's behind the doors and will always avoid the one with the prize, opens them all except door No. 777,777. You'd switch to that door pretty fast, wouldn't you?"

I cannot believe that all those Ph.D.s still got it wrong following that explanation.

The element that people who get confused by this seem to miss is that THE HOST KNOWS WHERE THE PRIZE IS. If the host didn't know, then his opening the doors wouldn't help (and in fact there's a good chance he would open the door concealing the prize).