Understanding the Monty Hall Problem

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621

Richard writes:

“And for whatever reason – forgetful host – passing vandal – act of God – we get to see the goat we already knew about – so what has changed?”

What has changed is we know which of the two doors has 0/3 chance of concealing a car and which door has 2/3 chance of concealing a car, pretty useful information for anyone who is not completely delusional.

Just because you know a winner or loser exists, does not mean knowing more information about the winner or loser, such as its identity or location, is of no use. That would be like saying, “I know one of these five people murdered the victim but knowing three who did not is of no interest, because I already knew there would be at least there innocent people.” Completely bonkers. Of course, knowing which door conceals a goat is similarly useful to any sane person.

Buxton then adds:

“…if the wise contestant does actually see a goat he or she is smart enough not to be influenced by it or by the manner of its exposure - the wise contestant is blind to the revealed goat - it’s a don’t-care goat.”\

I think you mean a “don’t care contestant”.

Anyway, if the goat is so irrelevant, next time you “play” the MHP, turn your back on all three doors and remain blind to the door which is shown to conceal a goat…

Behind your back, the host opens a door and reveals a goat and asks you if you want to stick with your initial selection of, say, Door 1 or switch to one of the other two doors. One door has a goat standing in its doorway but you do not know which one because you are the “wise contestant” who is “blind” to the location of the goat.

You’re not allowed to switch to both doors (as you like to imagine you are “effectively” doing) and you must nominate door 2 or door 3.

Whilst blind to the location of the goat (which you say “I knew existed” stomping your foot), how are you going to reliably pick the closed door with a 2/3 chance of concealing the car instead of the open goat door which has 0/3 chance of concealing the car?

Unless you know the location of the goat you cannot reliably switch to the closed door with the 2/3 chance of concealing the car and you might as well stick with your initial selection.

That is what being “blind to the goat” means and that is, also, the relevance and significance of knowing behind which door the goat was standing.

So on the one hand Richard says the goat is irrelevant and can be ignored, on the other, if Richard is actually blind to the goat because it is irrelevant, he is completely stuffed in terms of locating the closed door when Monty asks him if he would like to stick or switch to another door.

As for probabilities being expressed by Richard as being “2:1” . . . I give up.

Overall, another unbelievable contribution that quite simply takes the cake. Again.

622

Excellent post Jonathon.
" Completely bonkers." I like that :slight_smile: Sums Richard up to a tee.

623

Johthan says this… in 604…

The offer to switch is made when the only choice is between the first pick and the only other closed door.

And he’s absolutely correct - that’s when the host makes the offer to switch - after a useless goat - like a ship in the distance - hoves into view.

And that’s how he tricks you like the best stage magician - his clever timing - he opens a door and reveals nothing of significance then he makes the offer to swap. Wow! So simple.

Some people think it’s a one-for-one swap. They have to imagine either a 50/50 scenario or the mystical transfer of a 1/3 chance to a specific door

Timing is the key - had he done it the other way round - make the swap offer and then show the goat - it would be easier for the contestant to sus him out.

The order in which Monty does things is the key - he confuses the simple-minded with his timing.

And as for that other chap - Mr Saltmarsh - we are not the same person but we do have two things in common - Firstly both of us understand the MHP and secondly neither of us suffers fools - which is to say we try to enlighten those in darkness - but it’s hard work.

Yes - the offer to switch is made after Monty shows a goat - I actually knew that - I think everybody knows it - you did not need to point it out. It would be good for you to consider why Monty does things in the way he does - his timing.

624

Richard,

2 red balls and 1 white ball in a bag, remember ?
What’s the probability (chance, likelihood) that the ONE ball left in the bag, (after you’ve picked one and Monty has peeked in and removed a red ball) is the white ball?
You’ve refused to answer this question at least 3 times so far.

625

It was at post 321 (~300 posts ago!) that Richard first revealed his theory from an alternative universe when he expressed his, by now, quite intractable view:

“The goat-door is not eliminated from the mix just because we can see a goat- it still carries its original probability of a 1/3 chance of the car – similarly the door with the car behind it also has a 1/3 chance of the car.”

So at the end of an episode of Poirot, the villain has confessed and the two relieved innocent suspects say “Well we’ll be off now”, but Hercule Poirot stops them and says “Not so fast, monsieur et madam! Just because the real killer has confessed, there is still a 67% probability that one of you is the murderer! Do you think those probabilities change just because a confession has eliminated you both as suspects? Au contraire, mon amis, my special adviser, M. Buxton, assures me there is still a 67% chance one of you is the murderer as probabilities can never change!”

As Buxton explained by analogy:

“The probability of each door hiding the car is fixed by the number of doors and the number of prizes – no matter what happens this 1/3 chance per door – open or closed – car or goat – does not change.”

“The real choice that Monty offers is the original 1/3 chance door or the other two 1/3 chance doors together – he craftily shows a goat before making his offer – it makes no difference.”

Remember, I Richard Buxton, state:

“just because we can see a goat- it still carries its original probability of a 1/3 chance of the car”!

That’s right, even though you can see a goat, don’t be fooled because 33% of the time, it will turn out that you were mistaken and that goat was actually a Chevrolet. Easily done, it seems.

Reminds me of the apocryphal story about the doctor under cross-examination who is asked by counsel if it is POSSIBLE that the owner of the brain sitting on the doctor’s desk in a jar of formaldehyde could still be alive and the Dr replies “Well, I suppose he could be practising law somewhere.”

But now, a door that is shown to have a goat behind it has 0/3, zero, nada, no chance of concealing a car.

If Richard maintains otherwise, then he is dealing in principles derived from quantum mechanics or some altered state or universe he is yet to explain. But it is certainly not sound or earthly reasoning.

626

Palmer - your 610…

You’ve refused to answer this question at least 3 times so far.

Because it’s a silly question and I’m not your lap dog - woof - woof - woof.

If you find it too difficult ask a friend to help you with it… not me…

LISTEN UP
There are two red balls and One White ball - it’s a 2:1 arrangement - the chance of any random ball being white is 1/3.

God only knows why I bother with this rubbish…
God only knows why you bother with it…

Now - please - drop the balls.

627

And according to Richard’s “Theory of Preposterous Probability” when we see Monty remove a RED ball from the bag, it actually still has a 1/3 chance of being the WHITE ball.

Not to mention Richard thinks there’s no difference between whether Monty opens a goat door or a gust of wind blows one open.

Arrogance and willful ignorance - not the most attractive of personality traits.

628

Johnathan - your 611 - I particularly enjoyed the brain practicing law reply - very good - thank you…

You would have it that the 1/3 versus 2/3 arrangement is cancelled when the 2/3 part of it happens - carry on thinking that way.

I have it that the proportion of chance attaches to the door by virtue of the numbers involved rather than the result and does not change purely because the numbers do not change.

For me the explanation is an elegant simplicity (in my mind but perhaps not in yours) but for you - you present yourself with a special difficulty.

What happens to the 1/3 chance of the car that was enjoyed by the door that got opened - the open door with a goat?

You have it move by some force or natural law to the unaddressed door - I have never seen an explanation of this - people have it happening because they want it to happen - because in they believe the open door is 0/3 chance the car but they must preserve its previous 1/3 chance - so they declare the magic transfer.

It really is a magic transfer - why to that specific door? Why not in equal part to each of the other doors - much easier if you leave the 1/3 chance just where it began - attached to the goat-door with it’s 2/3 chance of a goat.

Let’s face it - if a closed door can have a 2:1 probability when it’s got a goat behind it - why can it not enjoy the same probability when it gets opened? - probability allows for either eventuality - it’s not 2:1 a goat but only if it’s got a goat behind it - that would be silly… that defies common sense.

Thinks again for the Brain thing - worthy of Rumpole…

629

@Richard said
"Because it’s a silly question". It’s the MHP, only an extremely simple-minded person wouldn’t recognise it as EXACTLY the same game.

And you won’t answer the question because you know the answer (2/3) exposes your ‘interpretation’ of probability for the piece of crap it is - just as you won’t do the experiment when a random door is opened because you know the answer in that scenario is 50/50. You’re a phony Richard, an uninformed blowhard .

“This will be my last here…”. you said. You can’t even get that right.

630

@Richard

“if a closed door can have a 2:1 probability when it’s got a goat behind it”

Just unbelievably stupid. When it’s got a goat behind it then it doesn’t have any probability, the outcome (it’s got a goat behind it) is known.
You really need to go and look up what “probability” means, and stop pretending you know what you’re talking about.

631

Let’s face it, Richard, you’re acting without logic, reason or common sense for that matter and a practical test would show the door with the goat does not have a 1/3 chance of concealing the car.

There is no “transfer” of probabilities by any “force” or “natural law”, “magical” or otherwise. Probabilities simply exist after being calculated with the all available information.

We do not even have to calculate the probability of the door with the goat concealing a car because it is not even a possibility. So we know it is 0/3. The outcome of opening that door is already known. And it is the probability of the outcome we are calculating.

Harking back to your imperfect horse racing analogy and use of terms such as 2:1 to express probabilities would seem to suggest that you, Richard Buxton, are confusing probabilities with odds. Yes, a horse has certain odds for the purposes of betting but once it has run last in the race the probability of the horse winning is zero and the probability of the winner winning is 100%. Nothing magical, just common bloody sense.

If, on the other hand, you, Richard Buxton, are correct and the door with the goat in the doorway retained the 1/3 chance of concealing the car which it had when the outcome remained unknown and the possibility of the door concealing a car still existed, you would win the car once every three times you switched to it. But that would not happen: if you chose that goat door one million times you would never win the car, whereas your 1/3 “hypothesis” implies that you would win the car 333,333 time, so there goes that abjectly stupid theory. Also because it makes absolutely no sense whatsoever.

And I agree with all of the others who wonder why you fail to provide cogent responses to the examples put up concerning balls in bags and murder suspects, for example.

Nor do you explain how you pick the door with the better odds if you have your back to the door and cannot see which door concealed the first goat to be revealed.

My only conclusion is that this has gotta be a piss-take, at least I hope it is for your sake. Otherwise, take it up with Ms vos Savant so she can set you straight. I am bereft of ways to explain the bleeding obvious.

The proposition that the probability of an outcome, such as a horse race or MHP, cannot change when factors affecting the probability of a particular outcome change, such as through the scratching of a horse or the revelation of a goat, is mindbogglingly absurd.

You cannot explain why you would never win the car if you chose the goat door every time and reconcile that outcome with your view that the door retains 1/3 probability of concealing the car even after the goat is revealed, can you?

Of course you can not.

632

Ah!

The penny has dropped in my mind - you chaps have been right all along - the quality of your argument and your strength of feeling has convinced me to alter my opinion…

Thank you so much - I now realise…

When the door gets opened and we see a goat that door and the goat behind it are completely eliminated from the equation - removed - entirely discounted as if they never existed.

Where before we had Three Doors - Two Goats and A Car - when the door and its goat get removed we’re left with Two Doors - One Goat and One car -
the original door having a 1/3 chance of the car and a 2/3 chance of the goat and the remaining door vice versa - because there’s just two doors - is that how it works?

Two doors 1/3 versus 2/3 in one door and 2/3 versus 1/3 in the other - and the opened door either 3/3 and 0/3 or if it’s gone away 0/3 and 0/3

By removing a door and a goat we dispense with one 2/3 chance of a goat but retain the 1/3 chance of the car - so creative - I see it all now… :slight_smile:

Thanks again…

633

‘Thanks again…’
… and thank you Richard. What shall we talk about now?

634

@Freddie Orrell- You are right, that is a better way of writing it out so that one cannot misinterpret the situation of the experiment.

635

OK, this is hilarious.
Buxton-Saltmarsh. Weird. I am actually myself, not some alter ego of Mr Buxton.

It would be good for those who are just doing a wind-up to come clean

Or perhaps you’re just in a mutual mass debate.

BTW, Seph, that’s the business. Shortest line I’ve seen to get it explained right.

636

Oh, and is there anyone who disagrees with the strategy? If so, I have a wee little gaming app, only 1$ a pop, I’ll play with you

637

“Two doors 1/3 versus 2/3 in one door and 2/3 versus 1/3 in the other – and the opened door either 3/3 and 0/3 or if it’s gone away 0/3 and 0/3”

At first I wondered if it was indeed Richard but the cogency is unmistakable…

638

I’m not sure I believe him.

639

It’s definitely a trap. Next time anyone replays the MHP, Buxton will spring from behind the first door opened with some kind of explosive, eliminating all remaining doors and the set simultaneously. He will then ask what the probability of that was.

Don’t believe it? Look at the clues which, as in Total Recall, such as when Quade notices his next victim, the psychiatrist, is perspiring in his “dream”, are quite subtle:

Clue #1. A Brit would have realised “penny dropped” was a metaphor and so would not have written “the penny has dropped in my mind”. This points to the author of the last post being some kind of humanoid, perhaps a cybernetic organism, giving a literal meaning to a metaphor as they tend to do.

Clue #2. What Buxton said in Clue #1 about the penny dropping was among the lasts things in the post I could understand. Instead of just adopting or adapting any of the reasoning given by others, Buxton provided an almost unintelligible explanation. Confoundingly, this points to the author being the real Buxton.

Clue #3. The post was relatively brief and not contrary, so far as I could tell.

Clue #4. Most of the post is so confusing, ambiguous and open to interpretation that it leaves sufficient room for Buxton to postulate another explanation without contradicting what he purportedly just said so the actual meaning of the tortured explanation could be debated for another 300 posts.

Clue #5. Saltmarsh has disappeared.

640

In the beginning we asked 'What is the sum of 2 and 2?'
Buxton and his Bux buddies answered 'It is 4!'
And it was good. They had the right answer. Then we discovered they were reasoning thusly: ‘Any sum of two numbers is 4’.
This assertion was challenged. In Comment 536 Jonathan pointed out that there were other sums of two numbers. Like 3 and 1. Or 4 and 0. The Bux buddies laughed and said 'See? Just proves us right. All sums of two numbers are 4!'
But then the devil appeared, using the name of Ross, and proposed in comment #570 that we should be able to find the sum of 2 and 3! The “all sums equal 4” theory was in serious trouble. But Buxton (and later his friends) simply said 'That’s irrelevant. Why even worry about 2 and 3. That’s not what we’re talking about!"

So here it is again. To all of you that think you get to switch to ‘all the other doors’. The same game, but with 4 doors. You pick one, Monty shows you a goat. Do you switch to ONE of the 2 remaining doors? If so, why? What are the odds of those 2 doors compared to the one you picked initially and how can you possibly claim that the reveal didn’t change the odds of the other closed doors? I’d love to hear it. I expect the same stony silence as before because it works just the same as the MHP and breaks the ‘all other doors’ logic.