Understanding the Monty Hall Problem

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741

Funny you should suggest this…

Richard, you’re not too old to learn - perhaps you could take an adult education course in mathematics :slight_smile:

I am on this course… starting 22 Jan next…

https://www.conted.ox.ac.uk/courses/details.php?id=R14P213MAW

Puzzles and pastimes can be both amazing and amusing. Mathematical puzzles often seem like magic. Mechanical examples include Rubik’s cube. Other topics include board and card games, game theory (hawks and doves) and Sudoku.

From the earliest times, the human race has been interested in puzzles for recreation and education. Early texts for teaching arithmetic such as the Rhind Papyrus were cast in the form of puzzles. More recent examples include the 15 puzzle and Chinese rings which appeared around 1880 and the Rubik cube which came a century later. Now many of us are besotted with Sudoku. No special knowledge of mathematics is required but a nimble and fertile mind would be heloful. Particular topics which will be explored in depth include voting systems as considered by Lewis Carroll, the rich group theory behind the Rubik cube (and how to solve it) and how to win at poker.

Here’s a good one- You complete the first half of a journey at 30 mph - what average speed do you need to drive at over the second half of the journey to make the overall average 40 mph?

Misters Bond & Saltmarsh will have no problems with this - not sure the rest of you.

742

It’s language Freddie… language…
Not mathematics…

and we establish that Monty’s revealed goat door has 0/3 chance of a car

Of course it’s not a car - dead-cert that it’s a goat - but the door HAD a chance of hiding a car - the door carries the chance attribute that a car might have been hidden there - you think in absolutes and allow no possibility of what might have been.

You may think me stupid but I’m not that stupid - I can tell the difference between a goat and a car. The door carries the attribute of being one thing or the other - the actual contents do not modify the attribution.

743

"The door carries the attribute of being one thing or the other"
Absolutely 100% incorrect. The door is ALWAYS a door. It’s what’s behind the door that we’re interested in. When we don’t know what’s behind the door we assess a probability of 1/3 to the object behind the door being a car, and a 2/3 probability to the object behind the door being a goat, When we discover what the object actually is ( a goat) it is nonsense to continue to assess the probability of that object being a car as 1/3
As i said before it’s the doors that are confusing you, much easier to consider 3 unknown objects in a single bag. The bag doesn’t have a probability of being anything, it has an attribute of being a bag which never changes. Probabilities are assigned to the ‘unknowns’ behind the doors not the doors themselves.

744

Richard wrote:

“Probability is to do with what might happen in the future – it can enable us to select the most favourable path to a preferred outcome.”

And the present, or outcome over n number of examples tells us whether the probability as forecast was accurate or, er, reasonable.

Your probability that the goat door has a 1/3 chance of concealing the car is neither reasonable nor accurate because it does not enable us to select the most favourable path to a preferred outcome."

If I followed your theory, I would be as successful picking the goat door as I would be picking the closed door I did not initially choose.

So how do you explain my failure to win the car whever I choose the goat door if, a you claim, it holds a 1/3 “chance probability” of concealing the car?

You are neither willing nor able to do so.

Nor can you explain how contestants who always switch from a door that yields 33 cars per hundred picks to the only other closed door will double their number of wins even though they switched from one door to just one other.

You are engaged on a philosophical quest when maths will do.

745

So how do you explain my failure to win the car whever I choose the goat door

It is not my place to explain your failures in life - you must solve those problems yourself and with the help of your therapist.

I wish you good luck…

As for balls in bags I have lost the plot and do not remember the scenario - neither do I have the motivation to find out.

746

What? 80 minus 30 = 50, so, er 50 mph? Really, Buxton, can you not also see that the probability of the other door closed concealing a car equals 1 minus probability of first door picked, ie 1 - 1/3 = 2/3?

Pretty obvious is it not?

And all borne out in the results when one runs the experiment.

How does Buxton explain the difference between his theory and the reality of the results of the experiment that show the first door has a 1/3 chance of concealing the car, the other closed door yields 67 wins per 100 and the goat door yields no cars?

747

‘Misters Bond & Saltmarsh will have no problems with this – not sure the rest of you.’

How do you know Bond is a ‘Mister’ … just like I imagine you are Richard?
60mph.

748

Not my failure, Buxton, your failure and everyone else’s, ie failure to win the car whenever they choose the goat door which you say should yeild 33 cars per 100 games.

How can this be, Buxton, if the goat door has a 1/3 chance of concealing the car?

Probability is simply an expression of the long-run average result when the experiment is run a sufficient number of times.

We are so at tangents to one another that I think you owe it to everyone here to do more to explain exactly what the hell you are talking about because none of it meshes with reality.

Clearly your views are not based on what happens in reality or founded in maths and are based purely on some philosophical view that an event should not be denied of its probability even if subsequent events would tend to do so.

Fair enough, too, but it does not alter incontrovertible facts such as probabilities being bound to reflect, or at least reasonably reflect, reality whereas your probabilities do not reflect reality whatsoever.

The main difference between Buxton and the rest of us seems to be that Buxton looks at how the world was when he walked on to the set of “Let’s Make a Deal” whereas nearly everyone else reassesses their view of the world and of probabilities after Monty has shown us which door has a 0/3 chance of concealing the car by revealing a goat.

Buxton sits their and says, “Well, that door could have concealed a car 10 seconds ago, so I say it still could!”

The other’s say “Hmmm, if that door cannot conceal a car, the 100% comprising the sum of all probabilities must now be split between the two remaining doors that can possibly conceal the car.”

Some people say these probabilities are divided 50/50 and others conclude, correctly, that these probabilities are divided 33/67.

But Buxton conclude that the total of probabilities for all doors capable of concealing the car equals 67%.

There is something more than maths at work here and it is along the lines of philosophy, but not reality, and Buxton has been very poor at explaining the gap between his theory (goat door will provide a car once very three games) and reality (goat door never yields a car).

Buxton cannot explain it and Buxton will never explain it because this disparity cannot be explained in a numerically provable manner.

749

"As for balls in bags I have lost the plot and do not remember the scenario"
You’ve certainly lost the plot, not to mention persistently refusing to address those tricky questions that your “theory” has no answer to.

"neither do I have the motivation to find out."
I’m sure that attitude will stand you in good stead when you undertake your maths course in January.

750

No Jonathan - it’s 60 mph for the overall average to be 40 mph… not 50 as you say.

This is more logic than Maths - Let me explain … if the first half is a distance of 30 miles it will take an hour - the overall distance is therefore 60 miles.

To make a 60 mile journey at 40 mph will take an hour and a half - but an hour has already passed - so the second 30 miles will need to be completed in half an hour. Hence 60 mph.

I forgive you your error and - knowing you as I do through this forum - would have expected that sort of off the cuff reply.

Experiments and trials to show proof are for babies - and those who have no confidence in their own abilities.

  1. Each door carries a ⅓ chance of the car no matter what lies behind it.

  2. A single door can be no better than ⅓ the car no matter what.

  3. A set of two doors can be no worse than ⅔ the car no matter what.

Does it get any simpler that this?

And btw seeing the goat does not help or inform anything.

751

Probability is simply an expression of the long-run average result when the experiment is run a sufficient number of times.

What! - you need to run experiments to determine probability?

Is this the level of our discussions? - Inability to assess the Three Doors - Two Goats and One Car scenario without experimentation - omg how dreary.

752

"Experiments and trials to show proof are for babies"
Really, I’m sure all scientists the world over will heartened that they can now abandon their laboratories and research projects. Of course since those same experiments and trials refute your delusional ideas it’s hardly surprising you adopt this pathetic posturing.

  1. Each door carries a ⅓ chance of the car no matter what lies behind it. FALSE

  2. A single door can be no better than ⅓ the car no matter what. FALSE

  3. A set of two doors can be no worse than ⅔ the car no matter what. FALSE

Does it get any simpler that this? Does it get any ‘wronger’ than this you mean surely?

And btw seeing the goat does not help or inform anything. FALSE

D Minus on that test I’m afraid Richard.

753

"What! - you need to run experiments to determine probability?"
Well you certainly should since you’re incapable of actually calculating it properly.

754

It was such a long time ago - I think Johnathan made this comment…

Nor can you explain how contestants who always switch from a door that yields 33 cars per hundred picks to the only other closed door will double their number of wins even though they switched from one door to just one other.

I can Jonathan - I can…

The contestant who switches away from a single door with a ⅓ chance of winning the car - moves to a set of two doors each with a ⅓ chance - that’s double in my world - as previously explained by C Bond - the switch is to a set of two doors - you may think of it as a single door if you want to. I consider it a set of two.

How do you explain how come the switch from ONE particular door to ONE other door might yield a double chance? A set of one to another set of one and it ends up with a ⅔ chance!?! What’s going on? What magic as been worked on the swap door to lift its rating…

755

Richard says “I can, I can” explain how switching from ONE door to ONE other door doubles a contestant’s chances but then answers like a politician:

Jonathan: “[please explain] even though they switched from ONE door to just ONE other.”

Richard: “I can Jonathan – I can…”

“The contestant who switches away from a single door … moves to a set of TWO doors[!]”

That’s called avoiding the question. It is also contrary to what Ms vos Savant says about the host asking if the contestant wants to “switch from door A to door B.” Reads a lot like giving up ONE door for ONE other door, not two.

As for these statements:

  1. Each door carries a ⅓ chance of the car no matter what lies behind it.

Here, Buxton uses the word s"no matter what" instead of “until we know otherwise”.

  1. A single door can be no better than ⅓ the car no matter what.

Again, “no matter what” should be replaced, this time with “all things being equal.”

  1. A set of two doors can be no worse than ⅔ the car no matter what.

Ditto.

Richard, imagine instead of the goat door being left open, every time the game is played the door is opened, by accident. just briefly enough for the contestant to see that it concealed the goat. The host then says: “Please open just ONE door and ONE door only. Either your own or ONE of the other two doors.”

As a contestant, what chance would Richard have of selecting and opening a single door that concealed a car if he used his knowledge about the location of one of the goats? About how many times would he locate the car by opening a single door if he used this information every time?

He dare not say but, if he uses the ill-gotten information about which door definitely conceals a goat (ie cheats), he can win the car around 67 times for every 100 attempts if he always opens switches to the ONE SINGLE DOOR that did not conceal the goat. But why would Richard cheat, by using this information, if it did not double his chances of winning the car by switching from ONE door to ONE other?

How did this ONE SINGLE DOOR yield double the returns of sticking with his own single door? And infinitely more than if he consistently mixed up the two other doors and erroneously chose the goat door as his ONE SINGLE DOOR?

Experiments are used to prove a theory. What practical experiment can Buxton devise that would prove a person who chose the goat door would ever win the car let alone enough times to justify a probability of 1/3 remaining after it has become a certainty that it conceals a goat?

Once again, there is none and he cannot do this. Probabilities are neither locked nor sacrosanct.

The other thing Richard cannot do is explain why, when one sticks with one’s first pick that is like picking one door only but when one switches to the other door, that is picking the other door plus the goat door.

Why does the goat door only come with the other door?

What is special about switching from door ‘A’ to door ‘B’ that without saying or doing anything else the goat door is included with door ‘B’?

Why is a decision to stick with door ‘A’ not similarly treated as sticking with door ‘A’ and the goat door?

I suppose it is. Door A = 1/3; goat door = 0/3, probability = 1/3.

Door B = 2/3; goat door = 0/3, probability = 2/3.

In either case, the inclusion of the goat door is unnecessary and does nothing to increase the chances of winning the car and, in any event, is contrary to the instructions of the host which are to either stick with door ‘A’ or switch to door ‘B’, no mention ever of switching from to bot other doors. Except in Richard’s mind.

Richard is proffering a false explanation for how a player who switches to just one other door wins 67% of the time because he cannot grasp the concept of the probability of the car being in one particular location changing as more information comes to hand.

If you were playing Russian roulette with another person and they blew their brains out in round one, would you still think you had a 1/2 chance of dying in a game to the death?

756

"The contestant who switches away from a single door with a ⅓ chance of winning the car - moves to a set of two doors each with a ⅓ chance"
No they don’t, they only get 1 door with a 2/3 chance. Changing the problem to fit your preposterous theory doesn’t cut it.

“As previously explained by C Bond” … the unopened door carries a 2/3 chance all by itself. Don’t put words in other people’s mouths Richard, that’s very naughty. and entirely classless.

I consider it a set of two." You’re considering a different problem then. Why don’t you stick to the one under discussion?

“How do you explain how come the switch from ONE particular door to ONE other door might yield a double chance? A set of one to another set of one and it ends up with a ⅔ chance!?! What’s going on? What magic as been worked on the swap door to lift its rating?”

It’s called MATHEMATICS not magic. I wouldn’t want to be the lecturer in your adult maths course for all the tea in China - it’s a suicide mission.

757

Why at comment 713 under Buxton’s name is there an erratum for C.Bond’s comment at 711?

Also, Buxton says:

“Probability is to do with what might happen in the future”

No, it is to do with calculating the likelihood of each POSSIBLE event, not what events might be possible.

For instance, after the goat door has been opened, that the car is won by a contestant who selects the open goat door is not a possible event. The probability of an impossible event is 0% or 0/3.

The probability of the first door picked concealing the goat is 1/3.

Therefore the probability of the only remaining door concealing the car must equal the total of all probabilities for all possible outcomes (being 1) minus the probability for all other possible outcomes (1/3 for door ‘A’), so 1 - 1/3 = 2/3.

Buxton could disprove that by showing that by sticking with Door ‘A’ his win rate did not eventually converge on 0.333 or by switching to Door ‘B’ his win rate did not eventually converge on 0.67. Or by switching to the Goat door and showing his win rate was greater than 0.

But Buxton is neither willing nor able to do so.

He is too busy covering for Bond.

758

OMG Russel’s Paradox and Set Theory is rattling my brain. Back To Monty for a break. The Russian Roulette analogy is interesting.

-the revolver we are using has only 3 chambers.
-two of the chambers are loaded, one chamber is empty.
-the cylinder/magazine is sealed so you can’t see into the chambers.
-you can spin the cylinder/magazine to any chamber you want when it is your turn.
-you go first.
-if you can’t decide then someone will pull the trigger for you, while aiming at your head.
-you win the game by staying alive.

At first you choose chamber L and you know some things:
-there is a 1/3 chance that chamber L is empty.
-you also know there is a 2/3 chance the other remaining chambers (M and N) have one empty chamber.

The person who loaded the gun and knows what chamber is empty stops you before you pull the trigger.
-chamber M is opened and you can see it is loaded (not empty).
-you are told you can choose another chamber if you want.

So chamber M is loaded.
-this does not affect what you know about the chances since your first selection of chamber L.
-you still know there is a 1/3 chance chamber L is empty.
-you still know there is a 2/3 chance the other remaining chambers have one empty chamber.
-knowing that chamber M is loaded allows you to eliminate chamber M from the other remaining chambers
-there is now only one chamber in the remaining others that you did not select. You can reasonably say that one chamber (chamber N) has a 2/3 chance of being empty… and switch.

The set that contains a 2/3 chance of having an empty chamber initially has 2 members. When a loaded chamber is revealed the set still has a 2/3 chance of having an empty chamber but now only has 1 member, the chamber you should select.

759

I thinks it is time, unless the penny has dropped once again, for Richard to explain in simple terms what is the effect or meaning of each probability he ascribes to a door.

As I and others have explained ad nauseum, we say the 0/3 probability for the goat door means if you choose that door 1000 times you will win the car zero times.

What does Richard say is the effect or meaning of the goat door having a probability of 1/3?

Similarly, for door ‘B’, I and others say its 2/3 probability means if you choose that door 1000 time you will win the car around 667 times.

What does Richard say the 1/3 probability he ascribes this door means?

And finally, the original door, we both say it has a 1/3 probability of concealing the car and so around 333 wins for every 1000 times it is chosen.

So, Richard, can you explain in practical terms the meaning of the 1/3 probabilities you ascribe to the goat door and the door ‘B’ - being the door not initially picked by the contestant?

760

"for Richard to explain in simple terms what is the effect or meaning of each probability he ascribes to a door."
You’re asking the impossible Jonathon, he can’t explain what he doesn’t understand.
Have the 2 goats and car all hidden in a row behind a curtain instead of separately behind three doors and Richard’s entire argument falls apart . Whatever he may say to the contrary he is effectively stating that a goat in plain sight has a 1/3 chance of being a car.
He seems to think that the doors themselves have some inherent probabilistic value that is unchanging, whereas in reality they are just doors, mere contrivances only there to conceal the goats and car from the contestant.

Seeing as I appear to be off Richard’s Xmas card list at the moment - I have no idea why :slight_smile: - I am in no doubt that, true to form, he’ll ignore the curtain scenario just as he has repeatedly ignored the 3 balls in a bag scenario.

You have far more patience than I do with your replies to this clown - you deserve a prize

(I pity the poor bastard who draws the short straw and ends up teaching his adult education maths course next year).