Ross, the answer is quite simple. Buxton and Saltmarsh pick one remaining door each. If it pays off, one of them has the car weekdays and the other at weekends.
@C.Bond
" He is a trickster who does not share the player’s best interests."
He’s an automaton, the only choice he gets to make is which goat door to open when you’vwe picked the car, and even then he’s obliged to make a random choice.
Whether the player “chooses the same door” or “sticks” he only has a 1 in 3 chance of winning - the terminology is irrelevant.
“The same statistical calculations can be applied to both choices.” No they can’t.
You seem to imply the choices are unconnected games, C Bond. Not so - they form a sequence within the same game. Imagine you and I were playing a card game, say, I don’t know, bridge or somesuch. After a game, before the cards were dealt for a new game we would expect them to be shuffled. But the contents of the doors are not ‘shuffled’ (re-populated) before the contestant’s second choice - they are choosing from the same hand as before, unchanged but with one having been revealed. It is the same game but with extra information (hence the first-choice door chance remaining 1/3).
Probabilities are not as rigid as the thinking of Buxton who seems to believe subsequent information cannot alter the probability of a particular outcome. He surmises that probabilities are a “forecast” and therefore fixed and incapably of being varied.
Previously, Buxton has said that the odds of a horse winning do not even change AFTER the race has been run. Buxton should be a billionaire! He could back the winner at 4-1 AFTER the race. But he’d be locked up if he insisted a bookie hold the odds of the winner the same as when the outcome was unknown.
Buxton must never have observed the constantly changing odds across a field of horses prior to the start of the race, after a “betting plunge”, for instance.
As for Buxton’s comment:
“What extra information? Between any two doors there will always be at least one goat – so what extra information is delivered when we’re shown what we know to be the case?”
The extra information is the LOCATION of the goat. But Buxton prefers to ignore this vital tidbit.
Without this information, there would be no benefit in switching to one of the two remaining doors.
And that is what Monty offers: the option of switching to one other door not two other doors.
Before the goat door is opened, the contestant does not know which of the two doors has the goat. They know at least one of them has a goat, but knowing which door has the goat is what makes switching so advantageous.
So, if I ask Buxton which of the two unchosen doors definitely conceals a goat before any door has been opened, he can only tell me with, at best 67% certainty, because he has incomplete information, and he would not bet his house on picking a door with a goat.
However, if I ask Buxton AFTER one of the doors has been opened, the additional information (location of one of the goats) allows him to nominate with 100% certainty which door conceals a goat, and he would bet his house on that because he can see WHICH door has the goat, and he did not know this before.
That is the additional information Buxton now has but which steadfastly he has chosen to ignore or has treated as irrelevant because, in violation of the rules of the MHP and the offer being made, Buxton was always going to choose BOTH doors. Fantastic stuff.
So the opening of the first door not only tells us which door has a 100% chance of being a goat door, it also tells us that the other door has a 67% chance of concealing a car.
But at comment 618 Buxton already seemed to accept all of this when he wrote:
“Ah! The penny has dropped in my mind –”
Only in his mind it seems…
It seem that Buxton has suffered a relapse and once again is choosing to ignore the rules and the utility of the first door being opened because he believes Monty is offering him the option of keeping his first door or choosing all remaining doors. No such offer is made and no such choice allowed under the MHP as stated by Marilyn vos Savant.
If Buxton wants to preface his comments with “let’s assume in the MHP the host offers the contestant the option of sticking with their first guess or swapping to the TWO other doors” there would not be a single person in all of Christendom who would be arguing 50/50.
It is precisely because the offer is pitched as being between one of just two doors that people get confused and are betrayed by their intuition to think this is a 50/50 proposition.
Under Buxton’s version, there is no problem . . . it is blindingly obvious to everyone that switching from one door to two would be advantageous. But in the actual MHP problem, the offer is made when only two doors remain as options, and that the choice is between one door and the other door is the source of the problem and the reason well-qualified mathematicians have fallen into the 50/50 trap. If it were not so, this thread would not exist. There would be no need to explain it was not a 50/50 proposition.
Does Buxton really think that so many people, including mathematicians, would have been tricked by the MHP if it truly were a choice between one door on the one hand and two doors on the other?
The only reason people are tricked is because the host in the MHP problem gives the contestant the option of choosing between one of TWO doors.
Buxton has taken his (personal but widely adopted) way of comprehending the MHP and decided this is how the game is actually played, and that, patently, is not the case. He is so convinced that his way of understanding the game is the game that he now regards the most useful piece of information (revelation of the goat) as being “nothing new” because he was always going to choose both doors when he will only be ever offered one. He can choose to stick with his door, he can choose the only other closed door or, if he is completely wacky and hell-bent on loss, choose the door with the bleating goat.
Dear Jonathan…
knowing which door has the goat is what makes switching so advantageous.
Really? You really believe that?
No dear chap - switching away from a single door with a 1/3 chance of the car to the alternative of two doors is what makes switching so advantageous.
Of no consequence that we get to see a goat that was certain to be there somewhere
Of no consequence which particular somewhere concealed the known goat.
Here’s your concept in action…
Goat revealed from behind the door on the left -good reason to switch
Goat revealed from behind the door on the right - good reason to switch
And here’s my concept
Who gives a shit where the goat is - we always switch…
The two concepts seem to have identical outcomes but yours seems over-complicated - as if some kind of voodoo magic applies…
It would be neat if you could elaborate on your statement…
knowing which door has the goat is what makes switching so advantageous…
you know - give a bit of detail - your thought processes that let you reach this creative principle - do say…
if you pick door one and turn your back on the doors and the host opens door 2 to reveal a goat, you cannot name the closed door to which you want to switch
Does it matter?
Is the contestant required to identify which door?
is it not sufficient to just say “switch” ?
Is this the level your naive mind operates at? Childish peekaboo…
Does any description of the MHP require the switch door to be specified?
If a door has been opened to reveal a goat - most people will disregard that opened door - you for instance - you think it no longer in play but at the same time of vital importance - but you’re unable to expand on the nature of the importance.
Me? I don’t care - a goat is just part of life’s rich pageant - I’m not concerned or affected by the goat - I switch regardless - You do too I suspect but you seem to make a big deal about it - the goat has great significance for you - as yet unexplained - but it looms large in your confusion.
Ah here’s Richard, being deliberately obtuse again… … not to mention (unavoidably) stupid, arrogant (his default condition), tediously repetitive and generally exhibiting his normal obnoxious personality.
“1/3 the car and 2/3 a goat – they represent this when they’re closed and when they’re opened”. Why doesn’t the game start with all 3 doors open then? You’ve got to laugh.
Why doesn’t the game start with all 3 doors open then? You’ve got to laugh.
Ho - ho - ho…
The joke’s on you Richard…you’re the clown here.
All right, Richard, it does not matter which door the goat is behind if all you are doing is electing to switch.
However, in the MHP as described by MvS, the contestant is unable to nominate to which door they will switch until the first goat has been revealed. Nor can the host nominate an alternative door to the contestant until this has happened. So the revelation of the goat and the identity of the remaining door is a vital part of the MHP and without which there would be no MHP.
There is no MHP if you insist that the host frames it, as you do, as a choice between your first door and the two other doors.
If that was how the problem was expressed, the solution to the MHP would not have caused so much, or any, controversy, including among experienced mathematicians.
Speaking of childish:
You insist that the expression of the MHP conform to your imagining of it.
I and others ask that the game be faithfully represented as an offer between Door ‘A’ (first pick) and Door ‘B’ (remaining door).
The latter version of the offer is consistent with the controversy it generated. Unlike your “all other doors option”, it does not make the many mathematicians who have got it wrong in the first place look even more inept, as no one would mistake your telling of the MHP as giving rise to a 50/50 proposition. Only the option of swapping from ONE door to ONE other door does that.
All right, Richard, it does not matter which door the goat is behind if all you are doing is electing to switch.
Correct - we agree absolutely…
the contestant is unable to nominate to which door they will switch until the first goat has been revealed.
The swap offer is not made prior to the goat being revealed so of course the contestant is unable to nominate which of two doors.
There is no MHP if you insist that the host frames it, as you do, as a choice between your first door and the two other doors.
I do not insist this - the host - Monty - is an automaton with little free choice - he follows a mental script - his only freedom of action is when the contestant first selects the door with the car behind it - Monty can open either of the other two doors to show a goat - he can make a random decision with a known outcome.
So the revelation of the goat and the identity of the remaining door is a vital part of the MHP and without which there would be no MHP.
I agree the first clause - the revelation of the goat injects confusion for both contestant and observers - without this act there would be no MHP - as for the identity of the remaining door being vital - the specific identity being vital - this door or that door - you have yet to show how this can be the case - it’s just a closed door with a 1/3 chance of the car.
One door on its own - a 1/3 chance of the car
Two doors together - any condition open or closed - 2/3 chance of the car.
Notice the chance of any door concealing a goat is 2/3 - this is because there are two goats.
For a single door to have a 2/3 chance of something - there needs to be two somethings.
And yes - even prominent mathematicians get their knickers in a twist over this oh-so-simple conundrum - so the opinion of a mathematician does not signify much - it is insufficient to say - trust me I’m a mathematician.
The goat-reveal is a simple, clever and wonderfully confusing theatrical trick.
Buxton says: “1/3 the car and 2/3 a goat – they represent this when they’re closed and when they’re opened”.
So according to Buxton if we start the game with all 3 doors open we still only have a 66.7 % chance of winning the car.
It takes a special kind of stupid to believe this, but then in the Alice in Wonderland world that Buxton inhabits maybe 66.7% represents a better chance than 100%. I’ve never studied make-believe mathematics so I can’t really say.
For a mathematician, Richard, you make a lot of spurious claims, such as:
“For a single door to have a 2/3 chance of something – there needs to be two somethings”
What about 67 goats and 33 cars? Would not that be a 2/3 chance also?
“Two doors together – any condition open or closed – 2/3 chance of the car.”
Not if both are open and have a goat standing in the doorway.
“the revelation of the goat injects confusion for both contestant and observers”
Confusion? Really? I thought it directed the contestant to the closed door with the 2/3 probability of concealing the car.
“it’s just a closed door with a 1/3 chance of the car.”
No, it’s a closed door with a 2/3 chance of concealing the car, as can be shown by playing the game enough times and always switching. On average, the closed door conceals the car around 67% of the time. So how can you say it has only a 1/3 chance of the car?
“The swap offer is not made prior to the goat being revealed so, of course, the contestant is unable to nominate which of two doors.”
Here you’re just being disingenuous. The reason the contestant cannot confidently nominate which of the two doors to pick is not because they have not been invited - even if they had been invited they could not pick a door with better odds than their first pick before the goat is revealed. However, once the goat is revealed, they can pick a door with a better chance concealing the car and they can do so whether they are invited to do so or not. So the sin qua non is not the lack of an invitation, it is the lack of the revelation of the goat (that prevents the contestant from nominating the door with the better chance of concealing the car).
And that is why I was saying the revelation of the goat is vital. You use sophistry to make it look like it is the host’s invitation that is vital, when clearly an invitation to swap without the revelation is useless. On the other hand, the revelation even without the host’s invitation still allows the contestant to know which door is more likely to conceal the car.
What about 67 goats and 33 cars? Would not that be a 2/3 chance also?
well almost a 2/3 chance but the MHP - of which we speak - has only 3 doors - let me put it in more simplistic terms for you…
In the Monty Hall problem when there are three doors - for any single door to have a 2/3 chance of anything there needs to be two anythings - you knew this - you’re just being silly - not becoming of an intelligent person
Not if both are open and have a goat standing in the doorway.
The door carries the forecast probability - the probability allows for either eventuality - the probability is not reset on the basis of one result or another - the probability talks of the future - how likely any particular outcome.
A closed door has a 2/3 chance of a goat - when opened to reveal a goat it still represents a 2/3 chance of a goat - similarly a 1/3 chance of the car.
To think otherwise is to confuse forecast with result - they’re independent of one another - they MUST be independent of one another - it’s language more than mathematics - the meaning of words that defeats you.
If a forecast could conceivably be affected by the eventual result then there could be no such thing as a forecast - only results.
I had to look up Sin Qua Non - not having heard it before in my 69 years - It’s actually Sine Qua Non - (without which not - absolutely essential)… I rewrite your passage thusly…
So the absolutely essential thing is not the lack of an invitation, it is the lack of the revelation of the goat
Surely the contestant requires an invitation to swap - is that not part of the MHP definition - the swap offer? Surely it would be rude to swap without the formal offer? The swap offer is essential.
As for the revelation of the goat that’s essential according to the MHP definition
(that prevents the contestant from nominating the door with the better chance of concealing the car).
I don’t think the contestant is required to nominate a door - he or she just elects to swap or not swap - in truth - as far as I recall - the MHP only asks about the relative chances of swapping or not swapping i.e at the point after the goat reveal - when the offer is made - the question is what is it best to do - not which door do you select - so all this confusion about what Monty or MvS says is bullshit - pure invention.
Even so - nobody can adequately explain how the revelation of a goat makes any difference at all - it’s just a goat where a goat ought to be - the (irony alert) big-deal goat.
So it’s like this…
Notice there is no nomination of doors - the specific door that gets opened is not mentioned because - because it doesn’t matter.
"A closed door has a 2/3 chance of a goat – when opened to reveal a goat it still represents a 2/3 chance of a goat – similarly a 1/3 chance of the car."
So if all 3 doors are open Richard what are your chances of picking the car? Most people would say it’s 100%, but you seem to think it remains at 66/7% which sounds like bollocks to me.
“the probability is not reset on the basis of one result or another” More bollocks, probabilities are constantly recalculated as more information is known (that’s why horse racing odds are constantly changing before AND during a race)
"So it’s like this…
If the probabalities don’t change whether a door is opened or closed, whether a goat is revealed or not, then Step2 is redundant and can simply be removed from the game. Yet more bollocks.
How does this truly bizarre “theory” of probability of yours translate to the 2 RED balls and 1 WHITE ball in a bag identical MHP? Does a RED ball begin with a 1/3 chance of being a WHITE ball? After Monty removes a RED ball from the bag does it still have … I mean does the 1 remaining ball in the bag have a 1/3 or 2/3 chance of being the WHITE ball?
It’s the doors that befuddle your feeble mind Richard, they’re only there as a means to conceal the prize and non-prizes, and could be replaced by a single bag. In your fantasy world though doors seem to be endowed with some magical property.
You don’t have a clue about what you’re talking about really … do you?
So if all 3 doors are open Richard what are your chances of picking the car? Most people would say it’s 100%, but you seem to think it remains at 66/7% which sounds like bollocks to me.
No Palmer - you’re being deliberately silly - I’m in the same camp as you and most people (all people) 100% chance the car - the open doors represent the results - the chance probabilities of each door remain as they were before being opened - I do not need to explain the relative chances - you’re smart enough to do it yourself.
There is no step 2 - the question is - is it better to swap or stick or does it make no difference? Most people get this right - I’m sure that you have no trouble selecting the best answer - what you seem to have difficulty with is the explanation of why it’s best to swap.
I shall not exercise my thought processes with silly variations - balls in bags - alternative MHPs with varying numbers of doors.
Why and How is it best to swap?
"There is no step 2 – the question is – is it better to swap or stick"
If there is no step 2 (i.e. no door is opened) which door are you going to swap to? You’ve got a choice of 2 doors.
“I shall not exercise my thought processes with … balls in bags” Why not? It’s the EXACT same problem. What’s the difference between 2 RED balls and 2 goats, 1 WHITE ball and 1 car? Explain how it’s different. Apply your “superior analysis” and come up with an answer. You can’t because your entire argument is based on a fallacy - that the probabilities in the MHP don’t change when a goat/RED ball is revealed.
You don’t know what you’re talking about, you’re a dullard of the first order.
The steps have not been defined or agreed as far as I’m concerned - some talk of the original selection as the first step and a subsequent selection as the second step - but the goat-reveal could be the second step - or the offer to swap - I can’t be doing with undefined steps.
In my view of the MHP certain things happen one after another - nobody holds up a board saying Step Two.
So there’s two red balls and one white ball behind three doors. Is that the puzzle? - Or are there two goats and a car in a bag - all jumbled up?
Why do you do this - modify a quite simple scenario of doors, goats and a car? I refuse to play your alternative games.
And while I have your attention - if you have to resort to cheap jibes, name calling and childish insults - it strongly suggests that you know your position unsupportable and that you’re probably trying to save face -conducting a loss reduction exercise - best to just admit the truth - don’t you think?
Good morning Richard. I would be fascinated to learn your response to the following question:
‘Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?’
Freddie - you’re very naughty - you dream up strange Monty-like variations for reasons I don’t understand - but this time I’ll humour you…
This is not the MHP and has no bearing on it
I pick door 1 - the host offers the chance to have another specific door.
All doors have the same chance probability as one another - each 1/3 the car and 2/3 a goat - door one and door two are both closed and each has the same chance probability of the car - 1/3.
How can it make any difference at all? - doors 1 - 2 & 3 have identical chances.
I seriously puzzle over Monty’s motivation here - I puzzle even more your motivation in asking the question - you play riddles and you lack openness.
I assume of course that Monty does not wink, nod or gesture with his elbow towards any particular door - he remains impassive and inscrutable and has no ulterior motive.
It is not to my advantage to change.
Equally I am not disadvantaged if I do accept the offer.