Understanding the Monty Hall Problem

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581

Richard? Have they changed? Have they?

Of course they’ve not changed…
In the first game the chances of the three doors are equal at 1/3 versus 2/3
And when the player makes his or her selection that’s the deal…

Then you begin a second game in which you remove the selected door - as if no selection had ever been made - and require the player to select again - one from two - and THEN -patronizing or What? You ask the odds - as if you didn’t know.

This juvenile attempt at sophistry does not wash with me or anyone else - you are an empty vessel - echoing with negativity - have you always been this negative?

You select Door 1 - Your selection status for doors 1,2 & 3 is… respectively…
Yes - No - No

If you reject Door 1 then perforce the arrangement becomes No - Yes - Yes

Notice you’re swapping one door (door 1) for two doors (doors 2&3 together)

And guess what?
Two 1/3 chance doors together represent a combined 2/3 chance

And guess what again?
Yes - you’ve guessed it - congratulations - behind any two doors there will always be at least one goat!

The goat makes no difference whatsoever - Whoopee!
The train has arrived at the station…
Always swap - ignore the goat.

582

"Then you begin a second game in which you remove the selected door – as if no selection had ever been made – and require the player to select again – "

Some progress here Richard - perhaps the MHP can be thought of as a two-game scenario in which each game could have different chances to the other.

"Of course they’ve not changed…"
You appear to argue that the combined chance of two doors, behind which are known to be a goat and a car, having a car to be chosen is 2/3. I would have put that at 100% and the chance of choosing the car at 50/50. Please note that in my bastardized version the location of the car is 100% certain to be behind one of the two ‘switch doors’ (both of which are closed) while in the MHP the location of the car is only 2/3 certain to be behind the two ‘switch’ doors (one of which is open showing a goat). Are you certain 2/3 is correct for the remaining two doors in the bastardized version?

583

Ok Richard, the thing you seem to be missing about the MHP is that you only get to open one door when it’s time to win the car. Without Monty showing you a goat, all three doors have a 1/3 chance. Yes, we know there is a goat behind at least 1 of any 2 doors but we need his help. At no point do you get to select 2 doors to open - if you could do that, we wouldn’t need to bother Monty you could just pick two doors to start with.

Let me propose a new variation to demonstrate. The game is the same, but we add one more door with a goat. 3 Doors with goats, 1 door with a car. You pick an initial door, Monty shows you a goat behind one of the 3 other doors. Should you keep your door or pick a new one from the remaining 2 closed doors? What are the odds?

584

“Let me propose a new variation to demonstrate. The game is the same, but we add one more door with a goat.”

Jonathan already pretty much covered this in 536. Richard’s response is at 540.

Richard, two questions are currently tabled for you:
569. Are you certain 2/3 is correct for the remaining two doors in the bastardized version?
570. Should you keep your door or pick a new one from the remaining 2 closed doors? What are the odds?

Please don’t forget mine at 569.

585

Freddy,

My game is a little different, in 536 Monty opens all of the other doors and Richard thought of it as being allowed to choose a whole mess of doors. I’m trying to get him to understand that you only get to pick one door and the odds for that door change when Monty shows a goat. He’s going to be in real trouble with my example when he’s forced to explain switching to a new door as picking 1.5 doors instead of 1. :slight_smile:

586

Ross - you have no imagination - alcohol has damaged you…

If you reject a door - shall we say door 1 ? Then that leaves doors 2 & 3 - and - as you agree At Least one of them hides a goat.

How are we helped when Monty opens one of OUR DOORS to show OUR GOAT ?

Come on - think it through - Two Doors - Both of them Ours (the contestant’s) - there’s either one or two goats lurking behind them - and he shows us a goat - and - and - and ??? What difference if the goat comes from behind door 2 or door 3 ? - the red door or the blue door ? - the middle one or the one on the right ?- we see a goat - SO WHAT? How are we better informed?

And you may find this difficult - - - - YOU may not to get to swap to two doors because you lack the imagination to have two doors - but I ALWAYS GET TWO DOORS because I have the foresight to make my swap BEFORE Monty shows my goat to me - I just don’t tell him - he prematurely opens my 1/3 chance door and shows a goat - as Homer Simpson would say Doh! Then he asks his swap question and I say , “Yes Please.”

When you’re a smart-arse like me you can do this - If you keep quiet about it nobody need ever know so nobody can object to two doors can they?

You would do well to believe nothing that people say and to work things through for yourself whenever you can - when the common heard are looking right - look left. When they look up - look behind you… Check out 558 again - start with an empty mind…

Here are three little gifts for you…

  1. A Giraffe has a long neck so it can drink - the long legs are to reach high into trees.

  2. A vacuum cleaner uses less power when the hose is blocked.

  3. In the Monty Hall game the goat provides no help - not if you swap to two doors - and nobody can prevent you from swapping to two doors - especially if you don’t tell them.

And one final thought - In your imagined view of the MHP - when he shows you the goat and asks if you want to swap or stick - how does the goat inform your decision? - billy-goat stick? nanny-goat swap? Just what are your thought processes when the goat turns up? - what would Homer Simpson say? How do you decide? - on what basis? - how are you better informed? What is the effect of the alleged filtering?

Answer on the back of a postage stamp please…

587

There you go Ross, I was sure you’d underestimated him.

588

“YOU may not to get to swap to two doors because you lack the imagination to have two doors”

The rules of the game prevent me from using my imagination in that way. You only get to open one door. That’s why you need Monty to open the other one for you and drop its odds of hiding a car to 0. He’s showing you which door not to pick and thus altering the odds for the other door.

“how does the goat inform your decision? … how are you better informed? What is the effect of the alleged filtering?”

All of these questions will be answered for you by considering my 4 door example game where Monty only shows 1 goat. Now you must choose a single door. How does your “smart-arse” system handle that? Again, what are the odds in this 4 door game?

589

As George Bush the younger said - “They misunderestimated me.”

Rules schmooles - if I keep quiet about it nobody knows.

You actually get two doors - Monty opens one of them for you…

590

Richard, sorry to poke you but, please, may I request your answer to 569: “Are you certain 2/3 is correct for the remaining two doors in the bastardized version?” (I just wanted to check that’s what you are actually saying in your response at 568).

591

At 465 Richard said:

“Remember the numerator is the count of how many cars there are…”

Then at 568 Richard said:

“Two 1/3 chance doors together represent a combined 2/3 chance [of concealing the car (singular].”

The two statements cannot be reconciled as there is only one car and yet the numerator is 2.

Anyway, Richard seems to be saying that the chances of him choosing a door with the car, when he breaks the rules of the MHP by secretly choosing two doors instead of one, are 2/3 and no one disagrees with this I do not think. Similarly, if Richard secretly chose all three doors, his chances of winning would be 100%.

So sneaky Richard, sniggers to himself and thinks “I have a 2/3 chance because in my mind, I am picking BOTH doors before Monty reveals the goat door. Clever me!”

Back in the real game, following the rules of the MHP, the “conformists” wait for “Monty” to reveal the goat door and quietly say to themselves:

“Our door has a 1/3 chance of concealing the car. There are two other doors. The open door has a 0/3 chance of concealing the car. Anyone can see that. A goat is not a car. The other door has a 2/3 chance of concealing the car. I will switch from my door with a 1/3 chance to the only other option, being the single door with a 2/3 chance of concealing the car.”

“Nooooooo”, says Richard. "You must have TWO doors to have a 2/3 chance. You see, the numerator demands it!’

Richard and the conformists play the game one more time but this time, before Monty can do anything, all doors bar the contestant’s chosen door are opened by an electrical fault. Each opened door is shown to have concealed a goat.

The conformists say, “Hmmm, the chances of our door concealing the car are 3/3. I will stick with my door.”

“Nooooooo”, screams Richard, “the other three doors must be included in order to get a 3/3 probability, and I secretly chose all three at the start of the game and that is the only way you can reach a probability of 3/3”."

“All doors have been taken into account, Richard. We took the total of all probabilities, being 1, and deducted the probabilities for each door with a goat, being 0/3 and 0/3. After some mental arithmetic (1 - 0/3 - 0/3) we arrived at our answer of 1, or 3/3.”

“Nooooooo” screeched, Richard, “that is not how I play the game.”

“Oh”, said the conformists, "sorry, we did not realise you were playing a completely different game where your secret thoughts were deemed relevant. If it helps you understand how a single door can have a 2/3 or 3/3 chance of concealing the car, knock yourself out. But don’t think for a minute that what you secretly decided to do at the start of the game alters the answer to the question of “What is the probability of each of the two remaining closed doors concealing the car?”

Richard says, "You mean like in a Formula 1 race when a driver retires and the commentator says “that has greatly improved so and so’s chances of winning the race and the World Championship?”

“Exactly, Richard, as new information comes to hand, the chances of each possible outcome vary.”

As Richard can see, the conformists are asking “What is the probability for each of the two remaining closed doors?” Richard seems to be asking a completely different question and refusing to accept that a single door can have a 67%, let alone a 100%, chance of concealing the car, as is shown, above, to be the case. New information, new probabilities.

So what question is Richard asking? Does it really matter if it is not the question posed by Marilyn vos Savant?

592

@Jonathon,
Richard frequently contradicts himself (sometimes within the one post), that’s what happens when you ‘make stuff up as you go along’ - you forget what BS you’ve said previously.

That’s the 2nd time he hasn’t replied to my 3 balls in a bag problem (identical to the MHP) - I think the 1 bag instead of 3 doors must be muddying his “clearer mind”, as well as showing up statements like “I contend that the opened door’s 1/3 car and 2/3 goat chances stay permanently with the door” for the absolute bollocks they are.

He even said that “it made no difference” if a ‘gust of wind’ or the contestant himself opened a door and revealed a goat - and then promptly changed the subject when shown that this was demonstrably incorrect.

He has no interest in learning and will never admit he is mistaken. He has a closed mind … and is a bit of a wanker tbh.

593

Saltmarsh & Buxton are not the same person…

Buxton has left the field of play - bored with the ramblings and petty challenges of closed minds - bored too by constantly having to repeat himself.

You can think the swap offer is to exchange one door for another single door - or - like me - you can think it’s a swap from one door to two doors (one already opened and showing a goat).

If you adopt my approach then there’s no confusion - but if you shoot for the one-for-one scenario then you have the insurmountable problem of having to explain how a door’s chances of hiding a car begin at 1/3 and miraculously change to 2/3 just because another door has been opened to reveal a goat.

It’s the stuff of fantasy - who has to see the goat to make the 1/3 chance leave one door and fly to another? What if the audience see the goat first and the contestant a few moments later? At what point does the 1/3 chance move about? What’s the trigger for it? Light bounces off a goat and falls on a person’s retina and probabilities change? Preposterous.

Do not attempt to answer any of these questions - they’re unanswerable.

Have an easy life - take the two for one option - you won’t regret it - apart from a few of the less-able contributors rudely trying to shout you down.

Pip-pip

this is absolutely and positively my last contribution here.
Good luck to you all…

594

This will be my last here…

So you’ve all had time to reflect about this revealed goat… Here’s the deal…

Brown goat = Swap / Otherwise Stick - or perhaps vice-versa

Billy-goat = Swap / Otherwise Stick - or perhaps vice-versa

Goat behind Blue door = Swap / Otherwise Stick - or perhaps vice-versa

Goat from behind door 2 = Swap / Otherwise Stick - or perhaps vice-versa

Goat from behind that door in the middle = Swap &c &c

Or could it be - do you think - the goat doesn’t really help - it doesn’t matter if we see it or not - or how it gets shown - might it be that the revealed goat has no significance whatsoever?

Might it be that the goat door is just one of the two alternatives on offer if we elect to Swap? You know - two (closed) doors each with a 1/3 chance of the car - at least one of which will be hiding a goat…

And those people who claim it’s helpful or an act of filtering are seriously misguided and misguiding - they claim usefulness but fail to show exactly how they’re better informed - the practical use they make of the revealed goat.

Now - put that in your bag of balls and shake it up…

Best wishes…

595

“This will be my last here…”

If that turns out to be true, it’ll be one of the few accurate statements you’ve made in the plethora of poppycock you’ve posted here.

596

Richard’s, like, “I’m swapping no matter what so what do I care whether it’s door 2 or door 3 that conceals the goat? I am going to abandon my first pick, door 1, no matter what.”

Well, the “practical use” of knowing the location of one of the goats that Richard wonders about is that it tells the contestant which of the other two doors to which to switch. That makes the revelation of the goat incredibly useful.

In fact, the revelation of the goat is the single most important and singularly useful thing that happens in the MHP.

Without the revelation of the goat, there would be no MHP to discuss. You would just have a one in three guess, unaltered by any additional information.

As it is, the host, Monty, unexpectedly or otherwise, opens one of the other two doors to reveal a goat. Monty then invites the contestant to abandon their first guess and switch to the only other closed door.

If Monty did not reveal the location of one of the goats and merely invited the contestant to change their guess to one of the other two doors, the contestant’s chances of winning the car would not improve from 1/3.

However, the location of one of the goats provides a compelling reason to switch to the other door, and that is the useful purpose to which the information about the location of one of the two goats is put.

Now, if that was the only loose end Richard needed to be tied up, perhaps, indeed, he will have no more to say on the matter.

597

Richard, the mission of Better Explained is to seek intuitive explanations for difficult concepts. The criteria for this are met by the ‘best of two doors’ solution to the MHP which you promote. It does indeed intuitively explain that it is ‘…to your advantage to switch’.

However others of us here claim to go further and explain that the chance of the remaining unopened door, in isolation, can be quantified. There is agreement that your two-door solution explains the MHP question, but to disprove the second claim other reasoning is required.

Can I invite you to step aside from the two-door solution to the first question and address the reasoning specific to the second question? I would be interested in your answers to the following:

Firstly my question:
‘577. “Are you certain 2/3 is correct for the remaining two doors in the bastardized version [569]?” (I just wanted to check that’s what you are actually saying in your response at 568).’

Reminder (555): There’s a game show with three doors hiding two goats and a car. The contestant chooses a door, and the host opens that very same door (and no others) revealing a goat. The host then invites them to switch their choice to one of the other doors. What would you say their chances of picking a car are now?

Secondly Ross’s question:
‘570. The game is the same, but we add one more door with a goat. 3 Doors with goats, 1 door with a car. You pick an initial door, Monty shows you a goat behind one of the 3 other doors. Should you keep your door or pick a new one from the remaining 2 closed doors? What are the odds?’

Note: In 571 I mistook this question for the one in 536, but it is different. I think your answer at 573 makes the same mistake and you answer as if Monty reveals one of your two remaining doors, but he does not.

May I suggest, for clarity, a separate post to answer each question. There is no need to repeat your two-door solution to the initial MHP question; it does not disprove that the chance of the remaining unopened door can be quantified in isolation.

598

There is a very simple way to express this problem. There are three ways this whole thing can go down:

C=Car
G1=Goat 1
G2=Goat 2

Way 1: I pick C. He reveals G1. If I switch, I get G2.
Way 2: I pick G1. He reveals G2. If I switch, I get C.
Way 3: I pick G2. He reveals G1. If I switch, I get C.

Thus, it would be advantageous to switch doors when the opportunity is presented. I hope this simplifies it for people like me who did not understand a thing you guys were saying in your computer simulation runs and abstract analogies.

599

Simple indeed. To avoid the sort of ambiguity that can lead to double-counting solutions that bring it back to 50/50, may I suggest re-wording Way 1 to read: ‘I pick C. He reveals one of the goats. If I switch, I get the other goat’.

600

The answer is there. but how to explain (apart from the math)
Well, I don’t care if Monty shows me a goat. Of course he does, that is what he is supposed to do. I know that. No extra information.
So his offer is ‘would you like to bet that the car is in your original choice’ or
’would you like to bet that the car is in one of the other two boxes’?
‘I’ll pay you if you are right in the first guess, or I’ll pay you if you are right in one of the other two’. Your choice.
One out of two is better than one out of one.
Showing a goat is irrelevant. Unless you want a goat.
It’s the offer that changes