BS: Monty Hall Problem

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 05 Nov 12 - 03:18 PM

I think you're playing with words without realizing it.

I mean "smart" in a the ordinarily understood, general way. To claim without qualification that pigeons are smarter than people (as some have claimed) is to use the word in a most inclusive sense. Ask a stratified random sample of a million native speakers of English whether pigeons, as a matter of fact, are smarter than people (yes or no), and you'll get nearly a million noes. Plus, quite as significantly, much laughter at the silliness of the question.

Those respondents would not be ill-informed. Nor would they be unfamiliar with the ordinary meaning of "smart." Say flatly, "Pigeons are smarter than people" and some would undoubtedly say, "Yes, in some ways," because there are certain human attitudes and practices they don't think very highly of. But I'm not speaking of "some ways." I'm speaking of general intelligence.

Pigeons are more efficient at learning to choose doors by trial and error in a setup like "Let's Make a Deal!" They're also better at finding their way home and at flying by flapping. Those abilities don't make them smarter than people, unless, like Humpty Dumpty, you wish to stipulate a special definition of "smart" (for example, "able to outperform people in certain non-intellectual tasks").

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 05 Nov 12 - 05:04 PM

"There's no doubt that there are some tasks pigeons are better than us at, such as finding your way home from a random place."

Buddy, I am here ta tell ya, that ain't true. I can pick my way outta anywhere. The ONLY reason I can't do it faster that a pigeon is on accounta I can't fly. If I could, I could knock the feathers off a pigeon in that scrap. Beat it home and have a pot on the boil for it's arrival. Better? I say nay.

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 05 Nov 12 - 05:09 PM

Well, like several other topics raised recently, this is one is distinctly off-thread, so we don't need to pursue it too far. I think the charge of playing with words "without realising it" is a little harsh, though. I have certainly not claimed pigeons are smarter than humans in the ordinary sense of the phrase ... But I do say there are several distinct 'ordinary senses' of the phrase, not just one.

You may be aware of the Turing Test for artifical intelligence. Roughly speaking it says if a machine and a human are given the same tasks and another human without direct observation can't say which is which, then we have no grounds for asserting the machine is not intelligent (within the tested realm). I would say that applies to animals versus human as much as machine versus human. Now think of some test for 'smartness in a specific context'. If the machine or animals outperforms the human it seems odd to me to be prepared to use the word 'outperform' but object to 'smarter in that context'.

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 05 Nov 12 - 07:06 PM

"Pigeons are more efficient at learning to choose doors by trial and error in a setup like 'Let's Make a Deal!'"

Q. What is the difference between a pigeon and a Hall contestant?

A. The pigeon gets to do the task over and over and over. Usually the set-up for pigeons or mice is to do something having only two choices, which are the same all the time, I believe. If there were three chioces could the pigeon learn to always switch? I don't know pigeon behavior; can make complex choices or only simple ones?.   

But a contestant only gets to play the game one time (tomorrow there is a different contestant), so he/she must play the best odds for getting the car. and there is a rational behavior for that. Switch!!!!

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 05 Nov 12 - 07:24 PM

And, let's get one thing straight. Pigeons cannot answer my arguement that each of the remaining two doors in the "second round" had the same initial probability of being a winner in the first round. Truth and outcome tables? Opening a goat door changes the rules and, ergo, changes the entire situation. I contend, yet again, that if one changes the rules and the situation, one changes the application of logical analysis.

Food for thought or pigeon feed?

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 06 Nov 12 - 02:01 AM

John: in the normal game a contestant played once, but in the experiment referred to the humans could play as often as they liked. Now, pigeons being greedy and students being ... students I guess it is quite likely the pigeons played more often, but we don't have the data.

Gnu: there's a difference between learning something and being able to explain or justify it. Ask any sportsman with a pair of lucky socks ...

Subject: RE: BS: Monty Hall Problem
From: GUEST,Jack Sprocket
Date: 06 Nov 12 - 02:02 AM

g = goat
c = car
x = open door
W = win without change
w = win with change
L = lose without change
l = lose with change.
C = contestant
P = presenter.

Three possibilities (123): ggc gcg cgg

ggc:

C chooses 1
P chooses 2 - gxc (forced choice)
    (i) C sticks - L
    (ii) C twists - w
C chooses 2
P chooses 2 - xgc (forced choice)
    (i) C sticks - L
    (ii) C twists - w
C chooses 3
(a) P chooses 1
    (i) C sticks - W
    (ii) C twists - l
(b) P chooses 2
    (i) C sticks - W
    (ii) C twists - l

Total outcomes for ggc: 8
Total wins if Stick (W): 2
Total wins if Twist (w): 2   

C must either Stick or Twist so the probability of win is 25%.

Repeat for the other two combinations, merely shifting the situation where P has a choice to one of the other positions.

Which is in line with what you might intuit: if the door C chooses is opened at stage 1, the probability would be 1/3. Needing to get two guesses in a row right must reduce the probability.

Now repeat the exercise where all 8 combinations of g and c are permitted.

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 06 Nov 12 - 10:45 AM

GUEST,Jack Sprocket -
Sorry, but your analysis and, therefore, your conclusion are totally wrong. See either or both:
Jim Dixon, 29 Oct @ 10:35am
TheSnail, 30 Oct @ 4:53pm (a little easier to follow)

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 06 Nov 12 - 02:20 PM

DMcG... "Ask any sportsman with a pair of lucky socks ... "

Can't agrue with that. I NEVER went huntin without wearin green socks. It MUSTA worked on accounta I got a whack of ruffed grouse in my huntin days.

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 06 Nov 12 - 04:04 PM

Guest,Lighter--
It is good that you don't design games for a living. They'd be boring. Game shows do want to give prizes, else no one would watch, but they don't want to give the best prize every time. If Monty does not know where the car is, he may expose the car on the first round...game over. Big whoop!

But he wants to generate excitement so he opens a door with a goat. It is important to understand that the door he exposes will not reveal the car, and it will not be the door you choose on that round. The door he shows is eliminated, and you go to the second part of the game. That is the game, and those are the rules. Monty has to give you information or there is no game, only pure chance.

Gnu--
We agree, I think, on the first round, your chance to of picking a goat is 2/3, picking the car is 1/3. On the second round, if you stay with your door, you still given yourself only a 1/3 overall because you took the same available door twice. But by making a change, you have now given yourself 2/3 chances to win the car; that is you chose two different of the three doors available in the game, thereby you have doubled your chance of finding the car. The only times this strategy fails (for the umpteenth time) is when you pick the car's door on the first round, but that chance was only 1/3.

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 06 Nov 12 - 04:16 PM

"The only times this strategy fails..."

Well, there ya go, eh? Change the situation, change the rules, change the choice.

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 06 Nov 12 - 04:58 PM

> Monty has to give you information or there is no game, only pure chance.

Correct.

However, if you don't appreciate the information (and the controversy here and elsewhere shows that few contestants could have), and switch on a whim, and boost your odds accidentally, that part *is* pure chance, at least as I understand chance. Or, equally, you could stay on a whim: a "whim" implies no consequential thought or deliberation.

And, win or lose, you would never know whether your decision had boosted your odds or not. All you would have is a goat or an automobile or Monty's proffered cash (which complicates your strategy to come away with something valuable, based on the principle of the bird in the hand).

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 07 Nov 12 - 02:27 AM

And, win or lose, you would never know whether your decision had boosted your odds or not.

I'm not sure follow you there, Lighter. By definition the odds are 'if you played the game oodles of times what proportion would you win'. You know that switching improves your odds every time. An individual game on the other hand wins or loses but does not affect the odds: that's true of any game without an 'end-stop' (For example, actual gambling games rarely allow you to continue once you have run out of cash/shirts)

Subject: RE: BS: Monty Hall Problem
From: Wolfgang
Date: 07 Nov 12 - 07:57 AM

Problems with conditioned probabilities are among the most difficult to grasp. A 90 minutes lecture is usually not enough to teach them sufficiently. Even one of my heroes, the great Martin Gardner, once got another variant ("one of my two children is a son") wrong on the pages of Sci. Am.

I have since long given up using intuition for such problems. I simply use Bayes' formula to arrive at the sometimes very surprising correct solution.

The verbal formulation of the problem is often underspecified (more formally, the "event space" is not stated unambiguously) and therefore, there is sometimes more than one correct solution. However, even if the problem is described in a way that has only one correct solution, often people don't see it.

Two easy extremes: (1) If Monty knows where the prize is and is always mean, and only gives you, after opening a door without the big prize, a new choice in the case your first choice happened to be correct, the probability of winning by switching is nil.
(2) If Monty knows where the prize is and is always benevolent, and only gives you, after opening a door without the big prize, a new choice in the case your first choice happened to be wrong, the probability of winning by switching is 1.

The two more common solutions, one of which is counterintuitive. (1) Monty knows where the prize is, and always (a) opens a door without the big prize and (b) gives you the choice to switch, then you should switch (counterintuitively), for the probability of winning then is 2/3.
(2) Monty himself doesn't know where the prize is and (a) always opens one of the two doors that were not your first choice and (b) in those cases in which the big prize was not behind the now open door always offers you the choice to switch, the probability of winning (whatever you do) is 50%, the intuitive solution.

Since you don't know what Monty knows you should switch unless you think he is mean, for even in the worst case, your probability of winning by switching does not increase.

Of course, in Monty's case, one can retrospectively find the correct solution the empirical way. Someone has done that and has looked at all available videos of the show. The clear result: the majority of the people has not switched, but in the majority of the shows switching would have been better. However, across all shows the winning probability for switchers was less than 2/3 (though of course higher than 50%) which clearly hints that sometimes Monty actually was mean spirited.

Learning by doing it repeatedly is not so successful, by the way, as someone has posted. After 50 repetitions (in the switching wins in 2/3 of the cases variant) still half of the participants in an experiment did not switch, though the difference between 2/3 and 1/2 should be obvious after so many repetitions.

I once found a way to make the majority of the participants switch in an experiment even at the first opportunity by increasing the number of "doors" to ten. Actually, the student in her diploma thesis used ten upside-down cups under only one of which was a prize. After the first choice, she opened eight of the ten cups but never the cup with the prize (the participants were fully informed about the procedure). Then she offered the participants to switch and most of them did (but only a tiny minority in the three cups control). The probability of winning for switchers in the ten cups condition was 9/10, of course. Though most of the participants did switch, when asked about the probability of winning by switching, even most of the switchers said stubbornly 50%, same as in the three cups control condition. (So much for intuition about why we do what we do.)

As I said above, most of the conditioned probability problems suffer from an insufficiently specified event space. A last example: If someone throws two dice in a dice box (you cannot look at the result) and then takes out one of the two dice from under the box showing a 6, what is the probability that the other dice also shows a 6? The intuitive response, 1/6, is only correct, if the other guy has not looked at the dices before taking one out from under the box, that is if the probability for taking a dice showing a 6 was not higher then chance.

If however, your opponent had a look under the dice box before taking the dice out that shows a 6 and will always show you that one of the two dices has a 6 whenever he can (like in a well known German game of dice, in which the double 6 is the second highest throw), then the probability that the other dice also shows a 6 is 1/11 (hint for those that do not use Bayes' theorem to find the solution: the probability of a double 6 is 1/36, the probability for any of the other five combination with one 6 is 1/18) . It pays to know that when playing this game.

Wolfgang

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 07 Nov 12 - 11:22 AM

Wolfgang--
You wrote, "The two more common solutions, one of which is counterintuitive. (1) Monty knows where the prize is, and always (a) opens a door without the big prize and (b) gives you the choice to switch, then you should switch (counterintuitively), for the probability of winning then is 2/3."

This is basically correct, but not complete...Monte will will never reveal your Goat Door under any circumstance, or the game is instantly over. Not an exciting game for a television audience. I don't know if that means Monte is mean or benevolent as he always acts same manner.

You also wrote that the actual game has been retrospectively analyzed, and the results showed that results did not match exactly the expected probabilities, no matter the strategy the contestants followed. This, of course, is to be expected because each game is a new event wherein even the least probable outcome can occur. Since there were hundreds of games played, it would be interesting to know how close outcomes came to expected probability throughout the run of the show.

Subject: RE: BS: Monty Hall Problem
From: Murray MacLeod
Date: 07 Nov 12 - 12:16 PM

I have to say that I am totally amazed by gnu's inability to grasp the essence of this problem.

I know from his postings on Mudcat over the years that he is intelligent, and has a well developed grasp of maths and engineering, so it is bizarre that an elementary probability problem such as the Monty Hall problem continues to elude him.

Then again, I got to grips with probability theory at a very young age, and since doing so have been aware that the vast majority of people have difficulty in reconciling the mathematical facts with their initial intuitive assessment.

You can test this by asking the average person, "How many people would you have to assemble in one room to make it more likely than not that at least two of them share the same birthday ?"

Once you understand factorials, the answer is devastatingly simple, but approached intuitively, the solution seems incredible.

( The answer is 23 btw)

Subject: RE: BS: Monty Hall Problem
From: Mysha
Date: 07 Nov 12 - 09:53 PM

Hi Jack,

Yes, I was wondering when someone would come up with that one.

You're right that there are four possible outcomes for a given distribution of a car and two goats, but they're not equally likely. The first two choices for the presenter were forced, so they will always occur for that particular choice of the contestant. The other two, however, are true choices, and both appear for the same choice of the contestant. This means that those latter two outcomes together are as likely as one of the other outcomes alone.

For the sake of creating equality, let's assume Mr. Hall flips a coin once the contestant chooses a door. If he has a real choice, for heads he'll take the leftmost choice, and for tails he'll take the rightmost choice, which is still a random choice as it's by coin toss. The fact that he tosses a coin splits the outcome of each forced choice up into two outcomes, but they remain the same forced choices as the flipping has no influence on them.

(I'll use your term "Twist", rather than the "switch" this thread otherwise uses.)

ggc:

C chooses 1
(a) P tosses: heads
    P chooses 2 - gxc (forced choice)
    (i) C sticks - L
    (ii) C twists - w
(b) P tosses: tails
    P chooses 2 - gxc (forced choice)
    (i) C sticks - L
    (ii) C twists - w

C chooses 2
(a) P tosses: heads
    P chooses 2 - xgc (forced choice)
    (i) C sticks - L
    (ii) C twists - w
(b) P tosses: tails
    P chooses 2 - xgc (forced choice)
    (i) C sticks - L
    (ii) C twists - w

C chooses 3
(a) P tosses: heads
    P chooses 1
    (i) C sticks - W
    (ii) C twists - l
(b) P tosses: tails
    P chooses 2
    (i) C sticks - W
    (ii) C twists - l


Total outcomes for ggc: 12
Total wins if Stick (W): 2, out of 6
Total wins if Twist (w): 4, out of 6

C must either Stick or Twist so the overall probability of win is 6 out of 12: 50%.
C's optimal choice is to Twist, whose probability of win is 4 out of 6: 67%.

Analogue for the other two starting positions.


Wolfgang: I suspect that the switchers in the ten cups experiment are a result of the last-man-standing effect, rather of people seeing how the cups not chosen would effect their odds.

Bye,
                                                                Mysha

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 08 Nov 12 - 08:30 AM

DMcG, I'm speaking of one's own odds of finding the car. You would only know that you had boosted your odds if you understood the calculation of the odds and the value of switching.

Most contestants could not have known this. Indeed, part of the fun of the game was the excitement generated by the belief that sticking or switching was based entirely on "intuition" or a straight 50/50 chance.

As I may have said before, part of what makes the 2/3 odds counterintuitive is that the two remaining doors are visible and present. The odds are an intangible mathematical abstraction, which in this case seem more "real" (to those unable to grasp them correctly) than the evidence of one's own untutored eyes. ("Two doors, 50/50.")

It's as though the odds reach down from some rarified plane of existence to affect those of us below - though, as you say, the "effect" is notional rather than material (they don't turn possible failure into certain success, they only nudge it in that direction).

...if I'm making myself clear.

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 08 Nov 12 - 11:57 AM

Lighter, you cannot boost your odds (or as I prefer, probability), you can only use them to advantage. Of course, to do that you must understand what the probabilities are. Or you can ignore them to your thereby disadvantaging yourself.

I'm not sure I understand what you are saying in paragraph two. I think you mean the math seems more UNREAL than the evidence of your own eyes. If so, I agree that the odds seem counter intuitive in an actual game situation (mostly because of the excitement and psychology of the moment) causing a contestant to make disadvantageous decisions. Also, they do not understand that the rules of the game clearly take it out of the realm of pure chance.

I can understand folks on stage having no, preparation not understanding the best strategy. What amazes me that gnu and some others don't grasp the strategy even after the posting of several grids, and narrative descriptions of the best play which they can analyze at leisure. The only reason I can think of for this: Since the best strategy dos not guarantee a win, it seems no better than any other strategy. But the numbers are the numbers, and that makes those folks wrong.

Jack's analysis is wrong for at least two reasons. P will never choose the same door as C, which he indicates could happen ["C chooses 2/P chooses 2-gxc (forced choice)]. Before that, he states there are "Three possibilities (123): ggc gcg cgg" But each 'g' is a separate entity which needs to be accounted for. If he differentiates them by some means (for example G1, G2) he will easily see there are six possibilities for ordering the prizes behind the doors. If the premises are wrong, the analysis is wrong, the conclusion is wrong--GIGO.

And Mysha, who seemingly agrees with Jack (but not quite), is also wrong. He has made up his his own game where Hall/Host/Presenter must flip a coin. Since it is not THE game, it is moot.

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 08 Nov 12 - 12:19 PM

Well, Lighter you are getting into really difficult areas there, because you are moving out of the realm of mathematics and into psychology, judgement and ad-hoc generalisations. It is indeed interesting but includes such questions as why anyone ever buys a lottery ticket, when the odds of them winning are so minute, and how this compares to buying a raffle ticket for your fok club or favourite charity, when winning is pretty much irrelevant to your motivation. Complex and intriging but nothing to do with Monty, except insofar as that is an example

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 08 Nov 12 - 01:19 PM

Agreed.

The point is that the mathematical probability, when not truly obvious (e.g., head or tails on one toss, 1 in 2), can seem not only counter-intuitive but also almost uncanny.

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 08 Nov 12 - 03:58 PM

Murray... "I have to say that I am totally amazed by gnu's inability to grasp the essence of this problem."

Likewise, I am sure. You are talking about probabilities based on a simple ruse... offering two choices, the first of which DOES NOT COUNT and therefore CANNOT influence determination of subsequent probabilities. You can think it does, you can "prove" it does, but it simply cannot. Allow me, again... just one last time...

THERE ARE ONLY TWO DOORS! Now, IF Monty doesn't know where the car is, there ARE three doors. But, Monty KNOWS where the car is, so truth and probability tables and fancy theorems based on there being three doors (FALSE!) are inapplicable. You cannot apply conditions and analyses from the first choice to the second choice because they are inherently different... there are only two doors.

Anyone who thinks there are three doors should be locked in a dark room with a small black and white TV with poor reception and forced to watch EVERY Monty Hall car-goat segment ever aired over and over and over and...

Ya know, if I ever get a chance to win that car, I am gonna switch... and if I get the goat, I am gonna tie it up in front of yer house. Don't worry... there is only a 50/50 chance of that happening.

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 08 Nov 12 - 05:44 PM

Gnu, I feel your pain.

The key is that there's a distinction between the odds of where the car *is* (1 in 2 for either of the two final doors, regardless)

AND

the odds that you, the contestant, will *find* the car.

It's too easy for a non-mathematician (like me) to confuse the two. You're thinking of where the car *is* in the final setup. It's got to be behind one of the two doors (odds for either door, 1 in 2). But the real issue is the odds of your *finding* the car when there were three doors, Monty knowingly eliminated one, and you get to stick or switch. A more complicated situation (odds if you switch, 2 in 3).

Remember, the 2 in 3 probability is still no guarantee that you'll win the car. It isn't anything tangible like a door or a goat. You just have a better chance of finding the car if you switch.

It still feels weird, but I'm now convinced. It's almost as though something spooky must be migrating from your first choice to your second, but that's not what's happening at all. All that changes is the intangible mathematical relationship among the choices. You can't detect it without having worked out the possible results one by one. The mathematical relationship is real even if not detectable by straight observation.

Check out Jim Dixon's chart last week of all possible results. It will take a while to absorb all the permutations, but the 2 in 3 odds are right there.

Subject: RE: BS: Monty Hall Problem
From: TheSnail
Date: 08 Nov 12 - 06:18 PM

GUEST,Lighter

The key is that there's a distinction between the odds of where the car *is* (1 in 2 for either of the two final doors, regardless)

AND

the odds that you, the contestant, will *find* the car.

I think I'm going to cry.

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 08 Nov 12 - 06:27 PM

Since something I said earlier is likely to be dragged back out, remember I also pointed out that the phrase 'the probability where the car is' is an informal gloss, and is better expressed as 'the probability of finding the car given no information beyond that immediately observable'. It is not really possible to talk about the probability of where a car is independently of the information you have access to.

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 08 Nov 12 - 06:53 PM

Let me flesh that out with a different example. In dealing with probability outside mathematical notation, it is very common to omit the information bit. People will say that the odds of heads or tails is 50:50. In fact, that's wrong. What you need to say is (statement:) the odds are 50:50 (information:) if you are using a fair coin. Without the fair coin information, the odds are not clear. Suppose I show you a coin and, having told you it is fair, I then tell you the last 100 tosses were all heads. Then, if you accept my claim of fiarness the odds remain at 50:50. Now suppose I had told you I had doubts about the fairness of the coin and the last 100 were all heads. Then is is reasonable for you to also have doubts that the odds are truly 50:50 and you would be well advised to bet the 101st head turns up.

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 08 Nov 12 - 07:01 PM

"It is not really possible to talk about the probability of where a car is independently of the information you have access to."

Correct. Someone tell me that is incorrect and I'll tell you...

In the second round, the ONLY information you have is that it is behind ONE of TWO doors. There are only TWO doors. Monty has negated your "probabilities" by the elimination of one door... it ALWAYS was happens because it's a TV show! Your analyses, predictions, probabilities, and predictions are based on false assumptions because they do not apply to the ACTUAL situation in the second round.

Yeah, I am just a good ol Kent County boy. Yeah, I took philosophy at uni. Yeah, I hold a Master of Science degree in engineering. Yeah, I am ALWAYS a guy who will accept when he is wrong and apologize... even done it at times just to quell the waters, so to speak, but THERE ARE ONLY TWO DOORS.

Please... could ANY of you explain to me why you think there are three doors?

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 08 Nov 12 - 07:29 PM

gnu--
I find hard to believe that a person trained in philosophy who has a second level degree in engineering is not pulling our collective chains by stubbornly asserting that this is a two door problem.

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 08 Nov 12 - 08:21 PM

The two concepts "where it physically is " and "how likely I am to find it" seem to be sufficiently distinct. If Snail can stop crying, maybe he can explain why no difference exists.

It's a car, not an electron.

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 08 Nov 12 - 10:51 PM

Gnu asks, "Please... could ANY of you explain to me why you think there are three doors?"

I'll try. It is asserted on the first round that there are three doors...because there are.

Whether the car is won by the contestant (you) is determined by two separate, but connected actions. The first action is picking a door. Your chance of picking the door with the car is 1/3. Since there are two goats available, the chance of picking a goat is 2/3. You don't win the car on the first round if you chose that door, but neither do you lose if you didn't choose that door. By rule, you will never be shown the car's door; you will be shown a goat's door; neither will you be shown the goat's door you chose. The door exposed is then taken away. You need a second action to complete the game.

Nobody is asserting there are three doors on the second round. There are two doors left. These are the same two doors that were part of the available three doors in the first round. They are in the same relative position, and the prizes are still behind the doors they were behind in the first round. The prizes are always one goat and the car. [If the car were eliminated, you would have no chance to win the car, but Monty has showed you that it is still available, because he exposed one of the goats.]

The action you need to take in the second round is to choose a door. Your choice can be your original choice, in which case you have kept your chance of winning at 1/3, because you chose only one door out of the three that were available to you for the complete game. You have simply made the same choice twice! But if you avail yourself of the chance to switch, you have then chosen 2 out of the 3 doors that were available at the beginning of the game, thereby increasing your chances to snag a car to 2/3!!

BTW, you may have a degree in Philosophy, and a Masters in Engineering, but the scion of JotSC has a PhD in Philosophy, emphasis in Logic and Linguistics. He has read this explanation, and the grid of TheSnail; he agrees with both. Actually, he wonders why the grid is not sufficient, in and of itself, to convince you of the proper probabilities and outcomes.

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 08 Nov 12 - 11:13 PM

> But if you avail yourself of the chance to switch, you have then chosen 2 out of the 3 doors that were available at the beginning of the game, thereby increasing your chances to snag a car to 2/3!!

This is another counter-intuitive aspect for the following reason. If I hold two lottery tickets out of a total of three, my chances of winning are likewise 2 in 3. In that case, however, I have physical control over both tickets at once. I can see with my own eyes that I have two of the three chances. In Monty's game, by way of contrast, I am *giving up* my claim on the first door I select when I switch to another. It feels the same as tearing up a ticket.

Giving up a claim on the first door would thus seem by everyday reasoning to make a difference in the final calculation; but in fact it does not.

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 09 Nov 12 - 02:17 AM

Lighter --

Forget about things being counter-intuitive; that only means that you haven't considered all of the components of the problem.

When I said that you were choosing 2 of the 3 doors, that was over two rounds, and with the knowledge that the car was still available on the second round.

That is not what your lottery scenario implies, and it therefore has no analogy to the Hall Problem.

You have been given the components you need fortyteen different ways by narrative and by grid with the correct analysis. You seem to be trying to reinvent the wheel instead of trying to understand the Game under the rules followed in the Hall Problem.

There are six possible outcomes in the Hall problem. Four of those outcomes result in getting the car if you switch at round two. Two of those outcomes result in not getting the car. There is no way to change that. If you don't ever switch, you can only win in two outcomes. And if you stay or switch on a whim you only win half the time.

So the order of descending likelihood of winning the are are:
Always switch -- 2/3 wins
Alternately switch or stay -- 1/2 wins
Never switch -- 1/3 wins

I believe someone(s) has posted all of this during the last week in one form or another, and it is really easy to set-up you own grid to prove it to yourself.

Subject: RE: BS: Monty Hall Problem
From: TheSnail
Date: 09 Nov 12 - 06:33 AM

Lighter

The two concepts "where it physically is " and "how likely I am to find it" seem to be sufficiently distinct.

Quite right but that isn't what you said that reduced me to tears. What you said (with my emphasis) was

The key is that there's a distinction between the odds of where the car *is* (1 in 2 for either of the two final doors, regardless)

AND

the odds that you, the contestant, will *find* the car.

Physically, the car is behind one, and only one, of the doors. It is not fifty percent behind one and fifty percent behind the other. The trouble is, we don't know which. That's where the odds come in. From the information available, we can calculate the probabilities. For your late comer, there are two doors with a car behind one and a goat behind the other so the 50:50 option is the best he can do. But, we have more information than that so, if we analyse it properly, we can calculate more accurate probabilities. In this case, that the odds are 1/3 for the door the contestant chose and 2/3 for the one that Monty didn't open. The 50:50 odds have now gone.

In the first part of your distinction above, you seem to think that the odds have some sort of physical reality. They do not. They only exist in our heads as a result of the analysis we have made of the information.

As a bit of an aside. It might help you to think, not of the door that Monty chose to open, but of the one he chose not to open. In two situations out of three, depending on the contestants first choice, he chose not to open it because it had the car behind it.

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 09 Nov 12 - 08:45 AM

> Forget about things being counter-intuitive; that only means that you haven't considered all of the components of the problem.

It also means that the correct solution is inconsistent with everyday experience. Its counter-intuitiveness is the chief reason that the problem generates so much controversy. That makes it of interest.

For decision-makers, the psychology is just as important as the math. So the analogy to the lottery situation is entirely relevant. It partly explains why the Hall problem *is* so counter-intuitive that a vast number of people (according to Wikipedia) angrily refuse to believe the correct solution even after it's been explained to them.

That's a very unusual response, even when something as bizarre as advanced physics is being explained, Quantum mechanics makes most people shake their head and wonder how it can be: but they don't shout that the numbers are wrong and that the physicists are obviously idiots. I find that difference fascinating and possibly very important.

One reason for it is that Joe Blow has never peered into an atom but he certainly has had to make choices between two or three options.

> You seem to be trying to reinvent the wheel instead of trying to understand the Game under the rules followed in the Hall Problem.

I'm trying to grasp fully the implications of the wheel and of the rules themselves.

Jim Dixon's chart/grid/summary shows that the 2 in 3 outcome is correct. I've pointed this out several times.

But that doesn't make it any less counter-intuitive, for reasons I've already explained.

Those reasons are of interest.

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 09 Nov 12 - 03:48 PM

Lighter.. "The action you need to take in the second round is to choose a door."

Yup. And there are two doors available. ONE choice between TWO doors (options). The fact that there were three doors at the start simply cannot influence ONE choice between TWO options in the second round.

If you stick, that is choosing one option. If you switch, that is choosing the other option. Basic, nitty-gritty of philosophy... the base of all human analyses of logical thought... A OR B. Positive or negative. Night or day.

Where the arguements proffered fail MY logic is that they fail to equate switching and sticking as ONE choice between TWO options.

Oh... re me having a go? Having "fun" by sticking to my arguement... Nope, not a hope. I am quite serious. And, until someone shows me "different", I will stick. When anyone can prove me wrong, I shall buy them a pint and apologize profusely.

Then again, I bought a Looto Max (yes, typo intended) ticket today for the $40M prize tonight and that is quite illogical. >;-)

Subject: RE: BS: Monty Hall Problem
From: Nigel Parsons
Date: 09 Nov 12 - 04:42 PM

Gnu:
Yup. And there are two doors available. ONE choice between TWO doors (options). The fact that there were three doors at the start simply cannot influence ONE choice between TWO options in the second round.

Yes, you have a choice between two doors, but not two identical doors.

What differentiates them?
A) On has a car behind it the other has a goat.
Unfortunately, we cannot know for sure which is which (yet)
B) One of the doors was selected earlier, at which time we knew there was only a one in three chance of it being the one with the car.

If Monty had left you with a choice of three doors (the one he has shown to contain a goat is not removed) how would you decide?
Your choice is still 1/3 now, so does the car still have an equal chance of being behind each door?
Your previously chosen door still has a 1/3 chance of hiding the car.
You know that the door Monty opened has a 0% chance.
Is the chance if you switch 1/3 (the same as your chosen door)?
If so you have three options which only add up to 2/3.

Should the odds on both the remaining doors have increased, or has Monty given you enough information to increase the likelihood that the unchosen door hides the car?

Subject: RE: BS: Monty Hall Problem
From: GUEST,Rev Bayes
Date: 09 Nov 12 - 06:04 PM

>>Yup. And there are two doors available. ONE choice between TWO doors (options). The fact that there were three doors at the start simply cannot influence ONE choice between TWO options in the second round.

When you first pick a door, you have a 1/3 chance of picking the car.
Yes?

If I now open a goaty door, the probability is still 1/3 I have a car.
Yes?

So if I have two options, and the option I've got is 1/3, the other must be 2/3.
Yes?

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 09 Nov 12 - 06:05 PM

Nigel... "If Monty had left you with a choice of three doors (the one he has shown to contain a goat is not removed) how would you decide?"

Ummm... I guess I wouldn't pick the goat? I kinda thought that would be obvious, no? Did I miss something? Maybe I did, on accounta that's as far as I read your post. Ya kinda lost me there so maybe you can explain that before I read any further. Fair enough?

BTW... I may not entertain any more discussions regarding logic this eve as I have old friends dropping by. Alfred, Tommy, and the Three Amigos... me, moi and Jimmy Suis. We shall play tunes and sing as best we can and solve the world's problems. Too bad Monty Hall isn't joining as I would put his goat on the BBQ and make hime eat the whole fuckin thing.

So, I was at the gym today and a gorgeous young thing started working out next to me. I asked my trainer what machine I should use to try to impress her and he replied, "The ATM in the lobby."

You ain't wrong. I ain't wrong. I understand that. Few do.

No matter where this goes or ends up, I just want to emphasize, NO, I ain't shittin. I truly believe I am correct in my assumptions and my analysis. The ONLY thing that bothers me (and it bothers me a lot... A LOT!!!) is that some have posted that I am having a lark... teasing... whatever... being a... it's hard for me to even type it... a troll.

Again, sorry for the consternation... if I am wrong.

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 09 Nov 12 - 07:24 PM

"The fact that there were three doors at the start simply cannot influence ONE choice between TWO options in the second round."

Gnu, you can tell yourself that until the day you die, and you will be just as wrong on that day as you are now.

You either don't know the rules of the Hall game, or you are pretending not to know them. I have my suspicion which it is. Shame on us for humoring you.

At any rate, I think Monte Hall would get a kick if he knew we have devoted so much bandwidth discussing his long ago game

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 09 Nov 12 - 07:41 PM

"You either don't know the rules of the Hall game, or you are pretending not to know them."

Let me explain the rules as I see them... AGAIN!

Three doors. Contestant chooses one. Monty opens another door behind which is a goat. Monty asks contestant to pick either of the two doors that Monty has not opened.

Are those the rules? Yes or no?

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 09 Nov 12 - 08:03 PM

No.

For the reasons why, read some of the correct analyses posted over the past week.

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 10 Nov 12 - 02:51 AM

At any rate, I think Monte Hall would get a kick if he knew we have devoted so much bandwidth discussing his long ago game

I'm sure you are right. But as I see it, the issue isn't really the game, and has, I suppose, been what Lighter has been concerned about for the last third of the thread or so. It is how it comes about that we as a species are so poor when it comes to dealing with probabilities and since the whole of life is highly dependant on making such judgements to what extent we can improve of this. Now, it is of course possible in some circumstances to go away and think about it for a week and carry out formal analyses. But in most cases we have a second or two to make up our mind. Do we think anything can be done to improve our decision making in such areas? My guess, is that the only way is to become aware how fallible we are in such circumstances and try to hold our judgement very lightly, being prepared to drop them when we have time to think things through.

I suspect gnu is either teasing us or, despite his protestations otherwise, finds it difficult to change from his original opinion. For example, gnu has not responded to the post from BK Lick @ 01 Nov 12 - 04:17 AM as far as I can tell. That in its turn was a variation of Jim Dixon's post of 29 Oct 12 - 10:53 AM which also seems to have been overlooked.

Subject: RE: BS: Monty Hall Problem
From: BK Lick
Date: 10 Nov 12 - 04:05 AM

"Three doors. Contestant chooses one. Monty opens another door behind which is a goat. Monty asks contestant to pick either of the two doors that Monty has not opened.

"Are those the rules? Yes or no?"

Yes, those are the rules. I cannot fathom why JotSC said they are not.

Subject: RE: BS: Monty Hall Problem
From: GUEST,Lighter
Date: 10 Nov 12 - 08:38 AM

Thanks, DMcG. That's part of it. But beyond the deceptiveness of such situations is a second issue: why the correct solution is not just counter-intuitive to so many people, but also infuriating (and possibly a little scary).

Subject: RE: BS: Monty Hall Problem
From: John on the Sunset Coast
Date: 10 Nov 12 - 09:12 AM

"Yes, those are the rules. I cannot fathom why JotSC said they are not."

Perhaps my original answer was too hasty or too flippant. Perhaps I should have written, 'Yes, but...'. The "but" or "buts" have been noted several times in the past week.

Subject: RE: BS: Monty Hall Problem
From: BK Lick
Date: 10 Nov 12 - 11:00 AM

What gnu needs is clarification, not mystification.

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 10 Nov 12 - 11:29 AM

Why is it infuriating? All I can do is make wild guesses with not a hint of evidence. But with that caveat I would suggest it is because we set up all sorts of patterns in our brain/mind/suppositions and in most cases these work well. As they need for us to survive. Others are relatively little used so we don't mind to much as the are deeply trotton paths. For most of us whatever we know of quantum physics so we do not have any dependance on it. As a result if we are wrong it is not too painful. On the other hand 'which judgement is best?' Is the kind of decision we might make dozens of times of a day. So if our basis is shown to be faulty is is almost literally painful because we must doubt many decisions of our life.


That's my guess, I don't doubt others may have better ideas

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 10 Nov 12 - 11:37 AM

Apologies for all the typos. That's what happens when I use my phone 6or this :(

Subject: RE: BS: Monty Hall Problem
From: gnu
Date: 10 Nov 12 - 05:31 PM

DMcG... "I suspect gnu is either teasing us or, despite his protestations otherwise, finds it difficult to change from his original opinion. For example, gnu has not responded to the post from BK Lick @ 01 Nov 12 - 04:17 AM as far as I can tell.rom his original opinion."

Addressed, quite adequately and VERY truthfully I might add, except for BK Lick's post. I dunno why you think I am supposed to respond to that post. It seems obvious that he addressed it to you. Did you respond? If not, why not?

After all, this IS a discussion about logic and reason and truth. What you write is what you get.

BK... "What gnu needs is clarification, not mystification."

You gotta be shittin me.

JotSC, re his response re "rules".... "Perhaps I should have written, 'Yes, but...'. The "but" or "buts" have been noted several times in the past week."

No shit eh? "But" just don't cut the grass. Facts cuts the grass.

Those ARE the rules. PERIOD. FULL STOP. NObody but me seems to fully grasp this simple ruse. Some even think I am mystified and some cast aspersions upon my intelligence and, far worse, upon my character. Yup, I AM mystified. But it's not because *I* don't understand a simple parlour game invented far before that great Canuck, Monty Hall, was even born, in order to entertain children.

There are only two doors. Ask Monty.

Subject: RE: BS: Monty Hall Problem
From: DMcG
Date: 10 Nov 12 - 06:09 PM

I said: gnu has not responded to the post from BK Lick @ 01 Nov 12 - 04:17 AM as far as I can tell.

gnu said: I dunno why you think I am supposed to respond to that post. It seems obvious that he addressed it to you.



---

Well, since it starts with "Gnu is very persistent -- I like that in a bloke. Against all odds, I believe I may have devised a thought experiment that might actually convince him to reexamine his so firmly held conviction" and finishes with "Will Gnu still insist that either one of these cards has a 50-50 chance of being the Ace of Spades? Or will he see that the chance of his original card being the Ace of spades is only 1 in 52, while the chance of the other card being the Ace of Spades is 51 in 52?", it certainly seems to invite a response from 'gnu'. In fact if you look at the original it mentions 'gnu' four times, and sadly fails to mention DMcG at all.

So that's why I thought you might wish to respond. If you think it will remove ambiguity, I will invite you to respond. I have to say, though, that I do not expect you to answer the question about what you believe the odds to be in the card game BK Lick describes.