Understanding the Monty Hall Problem

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681

I agree Richard, the chance here remains 1/3 whichever door the contestant picks. And yes, I too assumed Monty neither winks, nods nor gestures. And of course as you say, this is not the MHP.

But it is nearly the MHP (it will not have escaped your attention that I took my words from the MHP and my only change was to omit certain words). What words would you say must be added to make this into the MHP? We appear to agree that the answer to the MHP is that it is advantageous to swap, so what is missing from my version to enable that conclusion to be reached? What essential step(s) must be added? Only what is essential mind, no need to add anything that is incidental or unnecessary. The sine qua non you might say.

What must be added Richard?

682

Does the word Patronising figure in your vocabulary?

What essential steps must be added?

  1. You pick a door
  2. Monty shows a goat behind one of the other two doors
  3. At this point you have the chance to swap and

  4. You’re asked is it better to swap or stick or does it make no difference

Note very well - a question concerning probability is asked - there is no actual offer to swap - just how might swapping change your chance of winning the car. - In all probability the car is a child’s toy and the goat is a line drawing - but no matter - we can imagine a real car and real goats.

On the subject of a door having a 2/3 chance of hiding a goat - and it gets opened to reveal - say - a goat - the most likely result - and my absolute assertion that even though we see a goat - the result is a goat - the door’s chance probability is 2/3 for a goat - this is the thing y’all scoff at - because y’all have limited imagination - too absolute in your view of the world.

Imagine a fair coin with a 50/50 heads or tails probability - it gets flipped and lands tails side up - Now - here’s the crunch question - you can see the tail side of the coin - and there’s no intention to flip it ever again - but what are its chances if it should be flipped - is it 50/50 or has it become 100% tails because we see the tail side.

And in the MHP when a 2/3 chance door gets opened to reveal a goat - is it a door with a 2/3 chance of a goat - or is it a door with a 100% chance of a goat?

Ponder this and get back to me - there is no need to grovel before me - but I think you should


© RB - December 2014
All rights reserved


683

The revelation of the goat must be added. The invitation is unnecessary as, as Richard does, we can play the game and swap in our mind to the 2/3 door remaining after the goat is revealed even if we are not asked. However, as Richard explained, without the goat being revealed, we cannot even usefully swap in our mind.

I do not know what “chance probabilities” to which Richard refers are or how they differ from probabilities, however I do know that the idea that Richard’s view that probabilities for events are fixed at the rates assigned when information was less complete are complete bollocks.

For instance, start of game the probability of a door concealing a car is 1/3 each, totally 1. Richard however says this does not change even after the goat is revealed.

How come then if someone entered the room after the goat was revealed and was told what had already gone down they would say the the probabilities of a car behind a particular door were:

Goat door: 0%
First door chosen: 33%
Other closed door: 67%

And how come, if you played the game 1000 times, on average you would win the car the following number of times depending on your strategy:

Pick the goat door: Win 0 cars
Stick with first door picked: Win 333 cars
Swap to only other closed door: Win 667 cars

The long run average proves the theory of probability.

If Richard were correct about the probabilities not varying from the time first made, a person who picked the goat door every time in the 1000 games would win the car 333 times.

The person who came in late with better information would state, quite absurdly, You will win the car 333 times if you always choose the goat.

What makes the first calculation of probability so sacrosanct that it is locked on place and cannot be updated to reflect the reality of the situation?

Sure the probability of the Challenger space Shuttle completing a successful mission did not stay at 99% or whatever rate after the launch vehicle exploded?

Would Richard be there comforting the families of the astronauts telling them there is still a 99% chance they will see their loved ones again?

684

Richard Buxton asks us to “Imagine a fair coin with a 50/50 heads or tails probability – it gets flipped and lands tails side up – Now – here’s the crunch question – you can see the tail side of the coin – and there’s no intention to flip it ever again – but what are its chances if it should be flipped – is it 50/50 or has it become 100% tails because we see the tail side.”

That is not even analogous with what you are saying about the door with the goat in the doorway.

The analogy would be you claiming that AFTER the coin has been flipped, you maintaining that even though it landed “heads”, there remains a 50/50 chance that someone who picked “tails” will win the toss even though everyone can see it has landed heads up.

In the same way, you say the door with the goat in the doorway still has a 1/3 probability of concealing the car.

Once the horse is bolted, the is zero chance of it not escaping.

Once anything has happened, there is zero chance of it not happening.

So not only is your reasoning wrong, the analogy you use to support your incorrect reasoning is wrong.

685

You said


it will not have escaped your attention that I took my words from the MHP and my only change was to omit certain words

No you didn’t do as you say - you invented the words - there is No Swap offer - the question is in the form -

Is it better to swap - or stick - or does it make no difference?

Monty doesn’t actually say anything - Monty is a figment - he has no voice.

And we all know it’s better to swap - the academic question is

WHY is it better to swap?

My response is -
Because swapping offers the combined chance of two doors together.

Other people think differently (wrongly) - they have to overcome the issue of the flying 1/3 chance that departs from the opened door and magically transfers itself - the whole 1/3 complete - to one particular door - which as if by a miracle suddenly becomes a door with a 2/3 chance of the car - and people actually believe this - that a single door can have a 2/3 chance of the car!

They are unable to explain the phenomenon of the flying 1/3 chance - when challenged to do so they respond that it has already been shown elsewhere - they avoid answering the difficult question - they need it to be so - so it must be so.

They have it this way

A) Selected door - 1/3 chance the car & 2/3 chance a goat
B) Unopened door 2/3 chance the car and therefore 1/3 chance a goat
C) Opened door showing a goat - too difficult to contemplate - eliminated

Which as you can clearly see - requires the flying 1/3 chance to move on the command of a revealed goat from C to B - for it all to work.

Sophistry - sheer sophistry.

686

’ – you invented the words – there is No Swap offer – ’

No I didn’t - from the Vos Savant Parade article:
'He says to you, “Do you want to pick door #2?”'
and from the head of this very thread:
‘Do you stick with door A (original guess) or switch to the other unopened door?’

You assert the impossible Richard. No matter what mechanism of chosing two doors together you may prefer, the MHP requires a swap to ONE door. You have repeatedly said that, no matter what the stage in the game, one door on its own has a 1/3 chance of the car. Switching from one door with 1/3 chance to another with 1/3 chance cannot give any advantage. So your solution does not support the single-door swap.

687

671: ‘And in the MHP when a 2/3 chance door gets opened to reveal a goat – is it a door with a 2/3 chance of a goat – or is it a door with a 100% chance of a goat?
Ponder this and get back to me – there is no need to grovel before me – but I think you should
’

It’s 100% once revealed.

688
  1. ‘For a single door to have a 2/3 chance of something – there needs to be two somethings.’

Let’s give Monty a daily show, one car game per show. On Monday, the car was behind door 1; Jonathan the contestant chose door 1 and opted not to switch when Monty opened door 2 revealing a goat. Jonathan won the car! On Tuesday the car ended up behind door 2. Freddie chose door 2, and when Monty opened door 3 he switched to door 1 and went home with a goat. On Wednesday, the car went behind door 1 again. Saltmarsh chose door 3 and switched to door 1 - another winner! On Thursday, with the car behind door 3, Nonamejohn chose door 2. Nonamejohn felt there was no point switching from one 50/50 chance to another - result: no car. Friday’s car was also behind door 3 and Palmer chose door 1. However he forgot he had done so and tossed tossed a coin which left him with door 1, holding a goat. For Saturday’s show special guest Richard’s choice was door 3, followed by a switch to door 2 and, yippee, there was the car!

So three switched, two of them becoming winners (2/3), while of the three who chose not to switch there was only one winner (1/3). The car was behind each door twice (2/6 or 1/3), and two contestants picked the car first time (2/6 or 1/3). An expected set of results. On each of the six days, a car and two goats were available to put behind the doors. Six times that week a car was put in position. So not just two somethings, plenty of somethings; in fact, enough cars (somethings) for the car to be behind the ‘switch’ door four times out of the six. That’s a 2/3 chance. There is only one ‘switch’ door each game; two games in three it will conceal a car. One door, 2/3 chance of car.

689
  1. ‘For a single door to have a 2/3 chance of something – there needs to be two somethings.’

That’s not two somethings each game Richard, it’s two somethings every three games for a 2/3 chance.

690

Freddie dear friend
 what are you going on about with this


That’s not two somethings each game Richard, it’s two somethings every three games for a 2/3 chance.


a bizarre utterance?

Please come down to from your imaginary world - there is no actual MHP - when Mr Hall was running his Make a Deal show he would offer to buy the chosen door from the contestant there was no Swap situation - i.e whatever Monty may or may not have said is of no consequence here


vos Savant poses the question that if there were such a swap offer - would it be advantageous to swap?

And we all accept that it is a good idea to swap should there ever be such a game and we were in the hot seat - the contentious issue - why we argue here is


Why and How is it best to swap - where does the 2/3 chance come from?

I maintain 1/3 from the closed door + 1/3 from the door with a goat - there will always be a door with a goat - and the goat-door had a 1/3 chance before it got opened - for me there is no difficulty.

For y’all you have a lot of work to do - some very difficult explaining - which is perhaps why nobody has attempted to explain the 1/3 transfer from one door to another.

To accept my theory a person needs to accept that a door with a goat on view has a 2/3 chance of hiding a goat even though we actually see a 100% goat - i.e before opening the door it’s a 2/3 chance a goat and just the same 2/3 chance when we see the actual goat - chances are dictated by the numbers - not by the contents. Contents can vary but forecast chances remain fixed - you know - a probability - a forecast of what might happen - a likelihood- not a statement of what has happened.

Clarity of language seems to escape some people. Chance Probability - what might happen - not what has happened.

So when people ramble on about what Monty actually says - and multiple games and ball in bags and different numbers of doors it’s just fantasy stuff
 make believe - the challenge is not which door you select but simply - why is it best to swap?

Clearly I accept my theory - others - many others do not - that’s their prerogative.

I can explain to my own satisfaction how we get a 2/3 chance - it seems to me that nobody has an alternative theory - they can only obfuscate.

691

Bullshit x 3, Buxton.

Probability is only of use because of its predictive value. If the probability (yours) assigned to an event does not reliably predict the event’s frequency it is of no use.

As for probabilities being fixed, nowhere in the literature is it written that the probability of an event occurring cannot be adjusted to reflect subsequent events, changes in dependencies or better information.

If the open door with a goat in the doorway had a 1/3 chance of concealing the car, as Buxton maintains, a contestant would win the car, on average, 33 times for every 100 times they switched to it.

However, as everyone knows, anyone who switched to the open goat door 100 times would win the car zero times. This is entirely consistent with the goat door having a 0% chance of concealing the car as every sane person seems to maintain, not the 1/3 chance Buxton maintains.

In other words, Buxton’s estimate of the probability of the open goat door concealing the car is completely useless because it is unreliable because it is completely and utterly wrong, as are you, Buxton.

If you want to keep your head in the stand (or up your fundament) and pretend that a bona fide goat has some chance of being a car, that is a matter for you. However, the irreconcilability of your theory with reality and with every principle of probability with which it collides is so manifest that it might be time to stop being so condescending, “dear chap”, pull your head out and take a good look around at what is actually happening in the real world and then place your theory where the sun don’t shine, cos it just doesn’t work, make sense or stand to reason.

692

Monty first asks you to choose one of three doors and you select ONE door. Monty then asks you if you want to change your choice and make a selection that includes choosing TWO doors, those not in your original choice. Of course, change from a 1 in 3 chance of winning to a 2 in 3 chance. It doesn’t matter that Monty reveals a goat. He will always be able to do so.

I am fully convinced now, the player must switch.
It seems more an exercise in rhetoric than probability, or a good mash of both.

693

Mr Bond wins two prizes - one for brevity and the other for being correct.

694

"the contentious issue – why we argue here is
"
It’s not contentious, everybody here (except you) knows the answer and you’re talking out of your arse as usual.

“where does the 2/3 chance come from?I maintain 1/3 from the closed door + 1/3 from the door with a goat” Yeah a door with a goat has a 1/3 chance of being a door with a car instead. Only in your imaginary world does that make any fucking sense at all.

"nobody has attempted to explain the 1/3 transfer from one door to another."
Everybody HAS explained it, you just refuse to listen to people who know infinitely more about the subject than you ever will.

“To accept my theory” . Probability theory would need to be rewritten.

" a door with a goat on view has a 2/3 chance of hiding a goat even though we actually see a 100% goat “
Reminds me of this quote from Alice in Wonderland 'When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it
to mean – neither more nor less.”

"Contents can vary but forecast chances remain fixed " More drivel from the dullard (‘dullard: a stupid or annoyingly ignorant person’, so not an insult at all but a particularly apt description based on your comments so far)

"So when people ramble on about 
 ball in bags"
You didn’t explain how the MHP with 2 goats and 1 car is different to the MHP with 2 red balls and 1 white ball. Is it the doors Richard, is it? If we put both the goats and the car in one great big bag would it become a completely different problem? Do tell Richard, we could all do with a laugh to relieve the tedium of reading your ridiculous posts.

" it seems to me that nobody has an alternative theory " If you believe that then your English comprehension skills are as poor as your maths skills. You’ll find plenty of (correct) explanations elsewhere on the web too, I guarantee you won’t find ONE though that agrees with your crackpot ideas.

695

Probability is only of use because of its predictive value. If the probability (yours) assigned to an event does not reliably predict the event’s frequency it is of no use.

But probability does not guarantee one result over another.

Probability is not reset to some other value because of a specific result.

Probability allows for either eventuality - in proportion!

696

@C Bond

" Monty then asks you if you want to change your choice and make a selection that includes choosing TWO doors"
What game is this?
Monty: "James, pick a door"
James: "I’ll pick #1 thank you Monty"
Monty: "Do you want to switch to doors 2 AND 3 James?“
James” "Oooh yes please Monty, that doubles my chance of winning!"
Monty: “OK, let’s open doors 2 and 3 then and see what you’ve won 
aah, 2 goats. Never mind James”

697

Almost correct Mr Eldritch


You forgot that he opens a door - just an oversight I’m sure - not a deliberate lapse of memory - it goes like this


Monty: “James, pick a door”
James: “I’ll pick #1 thank you Monty”

Monty - “Let me open a door and show you a goat. Look - a goat!”

Monty: “Do you want to switch to doors 2 AND 3 James?”
James” “Oooh yes please Monty, that doubles my chance of winning!”
Monty: “OK, let’s open the other door then- and see what you’ve won 

aah, 2 goats. Never mind James”

It seems that you now understand what’s happening - well done.

698

Richard, I note you steadfastly refuse to explain how you reconcile your 1/3 probability of winning the car if one choose the open goat door with your statement that the goat door holds a 1/3 chance of concealing the car.

In terms of proportions, the only proportion in which the goat door will provide the car is the proportion of 0:3, not 1:2 as you maintain.

No one said probility guaranteed any result over another, however there should be some prospect of an experiment proving your 1/3 theory for the goat door and yet there is none.

The only relevant experiment, ie running the game 100 times or more, will render a result of 0 cars from choosing the goat door whereas IF YOU WHERE CORRECT, the the car would be yielded around 33 times for every 100 times the door was chosen, not ZERO times as is obviously the case.

So riddle me that, Bozo (because until C. Bond and Saltmarsh came on the scene I refused to believe any two, let alone three, people could be so asinine).

So how do you end up with zero wins of the car if you pick the goat door in an infinite number of games if the probability of 1/3 you stand by has even a remote chance of being remotely correct?

699

My prediction: the probability of ever meeting Buxton, Saltmarsh and Bond all in the same room at the same time is zero.

700

My prediction: the probability of ever meeting Buxton, Saltmarsh and Bond all in the same room at the same time is zero.

Here we agree - I live in Reading UK - I have no idea where you and the other two hang out - but even if it we all lived in or visited the same place I would have no intention of social interaction.

And even if by some chance we did meet together - you would be denied entry and not be admitted into the smart-cookies’ club. You would have to whimper outside with your nosed pressed hard against the steamed-up window.