Monty Hall Problem... For Newbies

[marting]

Dunning-Kruger perhaps?

Steve S

LOL

No, I suspect that if you stripped the problem of the Monte Hall aspects and just stated it in a simple form there would a different response and one much more accurate. At least amongst people with reasonable cognitive capability and some exposure to probability. There is something more at work. Considering how long the problem has been extant I would be surprised if it hasn't been studied academically.

 

[This is The End]

Right, so you agree that at the outset, there is a 1/3 chance that it's behind Door A, a 1/3 chance that it's behind Door B, and a 1/3 chance that it's behind door C. You choose A, then Monty shows you a goat behind Door B. So how does that 1/3 chance that it's behind Door B change when he shows you a goat?

There's still a 1/3 chance that it's behind Door B.

EXCEPT... After you see the goat behind Door C, there is now a 1/2 chance that the car is behind Door A, and a 1/2 chance that it's behind Door B, so it doesn't matter whether you switch or stick.

As an individual player, who only gets one shot at this, your odds of becoming one of the contestants who wins the car don't change at any point during the game no matter what you or Monty do. The whole problem, as stated is a red herring.

What you are forgetting is that when he shows you a goat from one of the remaining two doors it is not only giving you new information about the one that he picks (obviously), it is giving you new information about the one that he doesn't pick as well (not so obvious).

 

[This is The End]

Or read anglolawyers post a few times. As others have pointed out, it is about as clear and succinct as you can get.

Yes but that explanation invokes multiple trials, which is not required. And which Prometheus specifically tries to exclude.

 

[GlennB]

LOL

No, I suspect that if you stripped the problem of the Monte Hall aspects and just stated it in a simple form there would a different response and one much more accurate. At least amongst people with reasonable cognitive capability and some exposure to probability. There is something more at work. Considering how long the problem has been extant I would be surprised if it hasn't been studied academically.

Mythbusters speculated that there was also an element of personal psychology in play, where the level of frustration experienced at changing your mind when you were right first time is greater than the pleasure gained by a succesful 'swap'.

That is, if you decide to keep the goat you'll just take it on the chin in a "that's life" way, but if you blow the car you had in your hands you totally curse yourself.

 

[Delvo]

I have read the whole thread. I still think it is 50/50

You choose a door he shows you a goat, he is going to show you a goat no matter what is behind your door.
Your next choice will always be 50/50
I think you are all over thinking this.

For those that still doubt this, here is a link to python code which you can execute in the browser that simulates this problem. The code should be readable even for people that don't program.

That is my account I am linking to, I promise there are not viruses.

http://codepad.org/oCYoujgj

For a demonstration that I think more people could see & follow, here's how to do it in a spreadsheet. I'm using OpenOffice, but the formulas in Excel are identical except maybe needing to replace semicolons with commas:

|A|B|C|D|E|F|G|H|I
1|Door with prize|Door you choose|Door he opens if mismatch|coin flip|Door he opens if match|Door he opens, all cases|Remaining door|Staying wins|Switching wins
2|=ROUNDUP(RAND()*3;0)|=ROUNDUP(RAND()*3;0)|=IF(A2=B2;"";6-A2-B2)|=IF(A2=B2;ROUND(RAND();0)*2-1;"")|=IF(A2=B2;(6-A2)/2+D2*((2-(2-A2)^2)/2);"")|=IF(A2=B2;E2;C2)|=6-B2-F2|=IF(A2=B2;1;"")|=IF(A2=G2;1;"")

Then you can just copy & paste row two down to the next several thousand rows and immediately get thousands of trial runs, then throw in a couple of cells with something like
"=SUM(H2:H10000)" and "=SUM(I2:I10000)" to add up how many times each one wins. The results are in the form of door numbers 1, 2, and 3 for columns A, B, C, E, F, and G, -1 or 1 for column D simulating a coin flip dictating the result for column E, and 1 for a win or nothing for a loss in columns H and I. And you get a new set of thousands of trials and automatic totals every time you have the spreadsheet recalculate.

How the formulas work, for anyone who might not be familiar with them (don't worry, they're pretty simple compared to how they might look at first if you haven't used them before, and they're useful to learn in general anyway... and I'm laying it out like this so you can see and verify for yourselves that this spreadsheet does why i say it does, instead of just trusting my computer gibberish on it):

First, columns A and B use RAND(), a random number generator, to randomly set door 1, 2, or 3 as the one with the prize and the one you choose at first. RAND() creates a number between 0 and 1 with a multiple digits after the decimal point, so we have to do two things to it to get what we want. First, it's multiplied by 3 so the range of possible numbers it might give us is from 0 to 3 instead of 0 to 1:
=RAND()*3
Then we need that rounded up to the nearest integer. There are ROUND(), ROUNDUP(), and ROUNDDOWN() functions, but ROUND() would give us 0, 1, 2, or 3, and ROUNDDOWN() would give us 0, 1, or 2. All three of these need two "arguments" inside the parentheses, separated by a semicolon in OpenOffice: first, what we want rounded off, and second, how many digits after the decimal point we want it rounded off to, which in this case is zero. So we tuck the thing we want rounded off, RAND()*3 from above, into it like this:
=ROUNDUP(RAND()*3;0)

Since both columns A and B are random, sometimes they'll match and sometimes they won't, and which one of those is the case will determine Monty's next action. So we'll need separate columns to handle each situation. (Or we could stuff it all into one giant multi-layered if-then formula with if-thens inside of other if-thens, but breaking it down like this keeps things easier to see while editing.)

The simpler case is if you didn't already choose the winning door, so we'll do that first, in column C. Monty has to open the only door that's left, the one you didn't choose that doesn't have the prize. What I have in the table above is an IF() formula, and those need three arguments in the parentheses: the condition that determines what's going to happen, then the output if it's true, then the output if it's false. In this case, the part we're really after is the output if A and B don't match, which I've made red below. It's a simple math formula using "A2" and "B2" to refer to the cells in columns A and B on row 2 (the two door numbers we've already randomly chosen), subtracting them from 6, because all three door numbers, 1+2+3, add up to 6, so whenever two are already indicated then 6 minus them indicates the other one. Another option for output from an IF() formula, which I've made blue below, is to use quotation marks to just have it write any text you want in one case or the other; you could have the cell say "I'm going to DisneyWorld" if A2=B2 and "Death to the opposition" if they're not equal. But if you put nothing at all in the quotation marks, the cell will look blank in one case or the other, which sometimes makes certain spreadsheets easier to read:
=IF(A2=B2;"";6-A2-B2)

If I didn't care about making the spreadsheet look neat and clean, I could have just used the red part above, and then it would give a visible result in all cases, whether the player had originally picked the right door or not, even though this column is really only for when (s)he didn't. We'll use IF() to exclude output we're not using in columns D, E, H, and I as well.

If the player picks the right door on the first try, Monty has to choose from the other two, so we need a mathematical representation of a random choice equivalent to a coin flip even though Monty doesn't flip a coin. For this, we need to go back to RAND(), but this time, the math we're applying to it causes the output to always be either 1 or -1: first, rounding off to 0 or 1, then doubling that so 0 stays 0 and 1 becomes 2, then subtracting 1:
=ROUND(RAND();0)*2-1

We'll use that difference in a later formula in another cell, so that cell will end up using 1 half of the time and -1 the other half of the time. Meanwhile, to complete the coin-flip formula, we'll tuck it into an IF() to eliminate output we won't use, since Monty's choice of two doors doesn't apply when the player doesn't choose the right door the first time:
=IF(A2=B2;ROUND(RAND();0)*2-1;"")

Next, Monty's actual choice of door. This is in another of those standard IF()s that we're using here to keep the cell clear when it's not applicable, so we only need to watch the details of one of the three arguments in the parentheses, which only has a few possible inputs & outputs. The door with the prize is the door the player chose. If that's #1, we need the output to say 2 or 3; if it's #2, we need the output to say 1 or 3; and if it's #3, we need 1 or 2. That means...

if starting at door "1" in cell A2: 2.5±0.5
if starting at door "2" in cell A2: 2±1
if starting at door "3" in cell A2: 1.5±0.5

You can get individual custom outputs from specific inputs like this by putting multiple RAND()s inside IF()s inside more IF()s, but it's structurally simpler to find a mathematical formula that yields the only kinds of output you need in all of the few possible input cases it could get (in this case, randomly using multiplication by either 1 or -1 from column D to take care of the "±", making Monty's choice between two doors to open random like a coin flip):
=(6-A2)/2+D2*((2-(2-A2)^2)/2)

Then it all fits into one IF() which refers back to only one RAND() in column D:
=IF(A2=B2;(6-A2)/2+D2*((2-(2-A2)^2)/2);"")

After that the formulas get simpler. In column F, we just display the result from either column C or column E, whichever is applicable, so we can have them all in the same place:
=IF(A2=B2;E2;C2)

(We could have had a single formula incorporating everything in C and E all along instead of taking three columns, but it would have looked like this...)
=IF(A2=B2; (6-A2)/2+D2*((2-(2-A2)^2)/2) ; 6-A2-B2 )

Now that we know not only which one the player chose but also which one Monty opened, that only leaves one other door for column G, the one the player has the option of switching to, using the same simple little mathematical stunt as in column C:
=6-B2-F2

And now we can find which door the player will find out is the one with the prize: the player's original choice, or the one (s)he could now switch to, in columns H and I:
=IF(A2=B2;1;"")
=IF(A2=G2;1;"")

That will put a "1" in whichever column would be the winning move in this case, and leave the other one empty. If you wanted numbers in ever cell of these columns, you could use a 0 instead of the quotation marks, and the SUM() formulas I mentioned earlier would still have the same results.

Here's a short example of the kind of output such a spreadsheet will generate, using just 30 trials instead of thousands:

Door with prize|Door you choose|Door he opens if mismatch|coin flip|Door he opens if match|Door he opens, all cases|Remaining door|Staying wins|Switching wins
3|2|1|||1|3||1
3|3||-1|1|1|2|1|
2|2||-1|1|1|3|1|
2|3|1|||1|2||1
1|2|3|||3|1||1
2|2||1|3|3|1|1|
1|3|2|||2|1||1
1|3|2|||2|1||1
3|1|2|||2|3||1
2|2||1|3|3|1|1|
2|2||1|3|3|1|1|
2|1|3|||3|2||1
2|3|1|||1|2||1
3|3||-1|1|1|2|1|
2|3|1|||1|2||1
2|1|3|||3|2||1
3|2|1|||1|3||1
3|1|2|||2|3||1
1|3|2|||2|1||1
2|2||-1|1|1|3|1|
1|2|3|||3|1||1
3|2|1|||1|3||1
3|2|1|||1|3||1
2|1|3|||3|2||1
1|1||1|3|3|2|1|
3|2|1|||1|3||1
3|1|2|||2|3||1
3|3||1|2|2|1|1|
1|2|3|||3|1||1
3|2|1|||1|3||1
||||||TOTALS:|9|21

So there you have it: no theorizing & potentially overthinking, just plain dong it and seeing the results. Figuring out how/why the results are the way they are can come later, but getting the actual results removes any doubt or dispute about what really works and doesn't work.

 

[Delvo]

To me, the pattern of empty cells in the table makes it simpler, more straightforward, and plainer to see than I even expected when I created the thing! It's quite clear that the only way that staying on your original choice can win is if your original choice was right all along, and that's obviously only going to be one third of the time. Anything else but that is a distraction.

 

[Meed]

Lower case j. More to the point, with the best will in the world, the journey analogy failed in its primary task because ABX and ABY are different journeys. The rationale for combining them is even weaker than in the actual problem.

You don't have to combine them. Try thinking of it this way:

Picked Door|Car Is Behind|Monty Opens|Change|Stick|Probability
X|X|Y|LOSE|WIN|1/18
X|X|Z|LOSE|WIN|1/18
X|Y|Z|WIN|LOSE|1/9
X|Z|Y|WIN|LOSE|1/9
Y|Y|Z|LOSE|WIN|1/18
Y|Y|X|LOSE|WIN|1/18
Y|X|Z|WIN|LOSE|1/9
Y|Z|X|WIN|LOSE|1/9
Z|Z|X|LOSE|WIN|1/18
Z|Z|Y|LOSE|WIN|1/18
Z|X|Y|WIN|LOSE|1/9
Z|Y|X|WIN|LOSE|1/9

To show why the probability of the first path is 1/18: Let's say X is picked 1/3 of the time and 1/3 of those times the car is actually behind X, and then 1/2 of the time Monty opens Y. We multiply 1/3 * 1/3 * 1/2 to get 1/18.

In the third path, 1/2 is replaced by 1, because Monty only has one door to choose from. So we get 1/3 * 1/3 * 1 = 1/9.

Now, add up all the probabilities of the paths in which changing wins (1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 = 2/3) and compare that to the combined probabilities in which sticking wins (1/18 + 1/18 + 1/18 + 1/18 + 1/18 + 1/18 = 1/3).

 

[jiggeryqua]

I'm not sure if I fall in the mathematician camp, but the failure to communicate is going both ways here. I have no idea what point you've been trying to make with your last several posts.

Well let's see. The one you quoted, where I apologised to Glenn for not caring about his attempt to explain the MH problem by using an analogy that just confused the issue? That was an apology to Glenn for...well, it was what it was and if you were truly confused I can only recommend you try reading just the exchanges between he and I (though you may need the entirety of my contribution, I may well have made an assumption that anyone interested enough to comment on my posts has actually read the thread, rather than just leaping into the middle and expecting each individual post to be comprehensible in isolation).

In the post before that I made my first attempt to point out to Glenn that an imperfect analogy about journeys is of no help (that's where I cared so little about the analogy (and consequently my post) that I made some trivial typos that Glenn ought to have been able to adjust for, if in a charitable mood). I also explained to another poster, who addressed me directly with a poor attempt at explaining the MH problem, that a) it didn't work, for the reasons stated and b) that my issue with it had long since been addressed, as was made clear several times earlier in the thread.

Did you really have difficulty with that one? It's in English. I get paid (and laid) for writing. You claim there's 'failure both ways', because you dont understand what I said, when I wasn't addressing you - which only shows your fundamental failure to understand communication at all. If you would like specific failures highlighted (I'm here to help ), we could start with:

I have no idea what point you've been trying to make with your last several posts.

If you've been trying to get some Unified Theory out of several disparate, multi-part posts to several different posters, there's your problem right there. Other than that, I recommend anyone should own a good dictionary and a grammar guide, but also read some of the better poets, playwrights and authors, whose exploration of the potential of words makes a valuable counterpoint to the dry precision of academic language.

I could go back for more of my posts and try to guess what you didn't understand, but it might be simpler (and politer, more courteous, less aggressive, divisive and unpleasant) if you were to instead say something like "I really didn't get what you posted [here], can you explain it better?". That you didn't is also a form of communication...

 

[Meed]

[snip]

As an individual player, who only gets one shot at this, your odds of becoming one of the contestants who wins the car don't change at any point during the game no matter what you or Monty do. The whole problem, as stated is a red herring.

Having only 1 shot doesn't change your odds. It's still 1/3 if you stick and 2/3 if you switch regardless of how many times you play. Just like if you roll a fair die your chance of rolling a 6 is 1/6 on any given roll, regardless of whether you just roll once or thousands of times.

 

[jiggeryqua]

Mythbusters speculated that there was also an element of personal psychology in play, where the level of frustration experienced at changing your mind when you were right first time is greater than the pleasure gained by a succesful 'swap'.

That is, if you decide to keep the goat you'll just take it on the chin in a "that's life" way, but if you blow the car you had in your hands you totally curse yourself.

I still haven't seen that Mythbusters ep. (I should, I have it on Netflix) but that explanation makes perfect sense to me. The odds are relatively meaningless when you only play once (there's a post a ways back listing the odds of having a run of freakish results that skew the odds). Whatever the truth about Irrelevant Goats, at heart there may or may not be a car behind your door. Even though it's twice as likely to be behind the other door, it still might be behind your door. Why change? Change might throw away a car, which as Glenn reports is a bigger kicker than just 'not guessing right'.

So, should you change your choice when Monty shows you a goat? The answer depends, apparantly, on whether you're a robot or a person, on whether you have a calculator or a heart. No, I take that back, it was harshly hyperbolic purely for effect. But would you be wrong not to change? No.

 

[Meed]

The answer depends, apparantly, on whether you're a robot or a person, on whether you have a calculator or a heart.

Which is something that it seems economists were awfully slow to pick up on. People dislike loss more than they like gain.

 

[Recovering Agnostic]

So, should you change your choice when Monty shows you a goat? The answer depends, apparantly, on whether you're a robot or a person, on whether you have a calculator or a heart. No, I take that back, it was harshly hyperbolic purely for effect. But would you be wrong not to change? No.

If you're going to start accepting a solution other than the one that maximises your probability of winning the car, what exactly have the last 7 pages been about?

 

[jiggeryqua]

If you're going to start accepting a solution other than the one that maximises your probability of winning the car, what exactly have the last 7 pages been about?

And we're back to communication. The last 7 pages have been about several things, including my never disputing the mathematical solution to the MH problem. I freely confessed that the solution never stuck, that conceptually I did not understand it - but I did not dispute it. I have appreciated the generosity of so many posters in attempting an explanation and have expressed my gratitude to those who framed the matter in a way that made sense to me. I have also endeavoured to help those posters whose communication skills are not the equal of their mathematical ability (though I don't recall any gratitude...)

The last seven pages have also been about failures of communication (and failures to understand the nature of communication, the nature of understanding and the nature of humanity).

Skipping ahead to page 7 and the post you quote, I am at a loss how to explain it better and tempted to settle for 'read it again' (if that weren't also just like shouting at foreigners). So I'll try again: Is it correct to say that changing your choice of door improves the odds of winning a car? Yes, emphatically so. But would you be wrong to not change? Again, no. Emphatically no.

The first question is about mathematics, the second question is about people - so you see, their answers are not contradictory, even though they're different and opposite.

ETA: furthermore, the MH problem does not demonstrate, prove or even suggest that one should change. It isn't a justification of avarice, it does not discover the objective values of 'a new car' or 'a crushing sense of disappointment'. It merely clarifies the odds of a certain result (and of its associated, but less likely, result). You may well, hypothetically and while in the mindset of solving a maths problem, assume that you (and any other human) must change because odds. Indeed, the answer to the question "If you want to improve your chances of winning a car, should you change?" is, was and always will be 'yes'. That may have been disputed by other posters, I'm not going back over 7 pages with a fine-toothed comb, but it was never disputed by me. But you only need to go back a few posts to clarify that the answer to "Is it wrong to stick?" is "no". That would only be the case when the odds of it being behind the other door were 100%, and while IANAM, we have just repeatedly demonstrated that they're not.

 

[Recovering Agnostic]

And we're back to communication. The last 7 pages have been about several things, including my never disputing the mathematical solution to the MH problem. I freely confessed that the solution never stuck, that conceptually I did not understand it - but I did not dispute it. I have appreciated the generosity of so many posters in attempting an explanation and have expressed my gratitude to those who framed the matter in a way that made sense to me. I have also endeavoured to help those posters whose communication skills are not the equal of their mathematical ability (though I don't recall any gratitude...)

Maybe they felt, as I did, that your manner was rude, boorish and arrogant, putting the blame for your lack of understanding on the shoulders of the people who were making an effort to explain it, rather than accepting the possibility that the problem could be at your end. Do you get many expressions of gratitude as a result of telling people their communication sucks?

Skipping ahead to page 7 and the post you quote, I am at a loss how to explain it better and tempted to settle for 'read it again' (if that weren't also just like shouting at foreigners). So I'll try again: Is it correct to say that changing your choice of door improves the odds of winning a car? Yes, emphatically so. But would you be wrong to change? Again, no. Emphatically no.

The first question is about mathematics, the second question is about people - so you see, their answers are not contradictory, even though they're different and opposite.

Charming as ever. The answers are not contradictory, by your terms, but the "people" question (which I understood the first time, thank you) is irrelevant to the question posed. The question is quite clearly a mathematical one of probability, with no room for psychological win/loss imbalances or anything of that kind. It's interesting to explore the reasons why it's hard to understand such a simple concept, but that doesn't make the wrong answer equally valid in its own way, unless you're going to assert that for some people, saying the earth is flat would not be wrong because that's their deeply held belief and the emotional cost of rejecting it would be too great.

Taking the thread as a whole, it looks like you've finally understood the answer, but can't bring yourself to stop quibbling in a search for reasons why you're right and everyone else is wrong. Just my opinion, but if I'm wrong, maybe you need to work on your communication.

 

[jiggeryqua]

Maybe they felt, as I did, that your manner was rude, boorish and arrogant, putting the blame for your lack of understanding on the shoulders of the people who were making an effort to explain it, rather than accepting the possibility that the problem could be at your end. Do you get many expressions of gratitude as a result of telling people their communication sucks?

Do you get much in the way of true satisfaction from picking fights on the internet?

I was happy to say that I did not (at the beginning of this thread) have a concrete understanding of the MH problem. I was equally happy to explore, and report, why I didn't understand certain explanations - I clarifed and clearly stated the particular stumbling blocks. I am the only person able to judge whether any given explanation works for me. I am not the only person able to say that any given explanation is unlikely to work for everyone, an idea that a lot of mathematicians found as hard to comprehend as my difficulty with the MH problem. I am certainly capable (and willing) to say when an explanation personally directed to me has wilfully ignored the points I'd previously clarified as being the conceptual stumbling blocks I faced.

To adopt, as you have, the notion that if someone doesn't understand what you say to them then they are at fault, is to display a profound ignorance of communication. Having something to say, and being able to type...there's a little more to it than that. "Your communication sucks" is a communication - but not a particularly good one, which is why I've not used it (despite your dishonest implication). I have said (and say again) that is saddening that mathematicians (and other learned types) so often have poor communication skills. That's a good communication, though someone or other is bound to get offended by it, especially on a forum on the interwebs, eh? They'd be mistaken in their offence, due to their poor communication skills (in this case, comprehesion - communication is always a two-way process and both ends need skills).

Charming as ever.

Pshaw, you didn't even quote the bit where I got a tad snippy. So, you've established that you can be even less charming, which is a tu quoque and an escalation. I can recommend some good communication primers. Do you know of any accessible maths works?

The answers are not contradictory, by your terms, but the "people" question (which I understood the first time, thank you)

I'll take your word for it, though what you posted strongly suggests you didn't understand. 'My' terms, by the way, are the terms you responded to. You use the word like it was 'my' reality, something that had no bearing on anything real. But it was about people (if you were following the exchange, or took the hint to read back a few posts, I wouldn't need to be typing now). People are real too. Harder to describe with mathematics, which I freely acknowledge is the closest we're ever likely to get to fundamental truth, but real and relevant all the same.

is irrelevant to the question posed.

It's tangential to the MH problem. From context, and from direct unequivocal vocabulary, it was not presented as an answer to the MH problem. Again, skills are needed at both ends of a communication.

The question is quite clearly a mathematical one of probability, with no room for psychological win/loss imbalances or anything of that kind.

'The' question? You mean the Monty Hall problem, which asks a question about probabilities? Yes, we're all up to speed on that, thank you. As an adjunct to that, a sidebar developed in conversation, relating to the 'psychological imbalances' (if you insist). Again, reading even just the last few posts would have made that clear. One cannot judge an ongoing communication by grabbing bits out of the middle at random and shouting angrily at them because you're thinking of something else. Well, no, obviously one can...it's possible. But I wouldn't. I'd look silly...

It's interesting to explore the reasons why it's hard to understand such a simple concept, but that doesn't make the wrong answer equally valid in its own way, unless you're going to assert that for some people, saying the earth is flat would not be wrong because that's their deeply held belief and the emotional cost of rejecting it would be too great.

Another poor analogy, though it does in some way serve your purpose. Alas, it's your purpose that's mistaken.

At no point did anyone claim that it was 'equally valid' to suggest the solution to the question "Is the probability of success in the MH problem improved by changing doors?" is both yes and no depending on whether you're a bit of a woo. It is encouraged round here to snipe at woo, and nobody much minds if anyone fires wildly as long as the noise scares off the nutters, but really, you're firing at shadows.

It is, however, correct (and not at all wooish) to note that the answer to one question is 'Yes' and the answer to a different question (that happens to be about part of the same situation) is 'No'.

Anything else I can help you with?

Taking the thread as a whole, it looks like you've finally understood the answer, but can't bring yourself to stop quibbling in a search for reasons why you're right and everyone else is wrong. Just my opinion, but if I'm wrong, maybe you need to work on your communication.

Which merely demonstrates you've either not read the thread as a whole, or wilfully failed to understand very clear posts.

 

[Roboramma]

But you only need to go back a few posts to clarify that the answer to "Is it wrong to stick?" is "no". That would only be the case when the odds of it being behind the other door were 100%, and while IANAM, we have just repeatedly demonstrated that they're not.

What does "is it wrong to stick?" even mean? Are we talking about a moral question here? I don't think there's any moral implication in the Monty Hall problem, so no, it's not morally wrong to stick.

If you are only concerned with cars and goats, and would prefer a car over a goat, then yes, switching is better than not switching.

If you are concerned with having a warm glow, and you get a warm glow sticking with your original choice, then it's possible that you should stick: if that warm glow is worth 1/3 as much as the difference between a car and a goat.

By the way, out of curiosity did you find my recent post in reply to yours to be clear?

 

[jiggeryqua]

What does "is it wrong to stick?" even mean? Are we talking about a moral question here? I don't think there's any moral implication in the Monty Hall problem, so no, it's not morally wrong to stick.

You're conflating two questions. Of course the answer to one is irrelevant to the other. There is no reason the MH problem should have any moral implication (we weren't discussing morality, nor even ethics, but yes, one of those wordy head-thing so-called 'soft' so-called sciences). There is no requirement for the other question to exactly mirror the MH problem (or it would fail to be a different question at all).

Oh, sorry, I was confused too - you took 'morality' from my use of the word 'wrong'. Fair enough, my lack of foresight is to blame. I didn't intend the word holding that spot to have moral implications. It would be 'wrong' as in 'incorrect' if your goal is to take one shot at glory. The odds, as has been discussed by evidently learned mathematicians here, relate to repeated iterations. We can take a punt on long odds if we want, sometimes they pay out. That's a fact too. You can change your pick and lose the car. That's a fact too. If we want to maximise the best result, we change. If we want to minimise the worst result, we stick.

Thank you for helping me clarify that.

If you are only concerned with cars and goats, and would prefer a car over a goat, then yes, switching is better than not switching.

If you are concerned with having a warm glow, and you get a warm glow sticking with your original choice, then it's possible that you should stick: if that warm glow is worth 1/3 as much as the difference between a car and a goat.

By the way, out of curiosity did you find my recent post in reply to yours to be clear?

You really weren't that confused, sorry. The comparative value is a good point too - what is the worth of the change, to the door that the undisputed and comprehensively explained MH odds apply? The odds do not dictate the action of the player, who is only a foolish woo (regardless of his ultimate choice) if he plays on the misunderstanding that his odds are anything other than 1 in 3 to stick (or believes Thor will intervene on his behalf, or such other nonsense as may prevail in his noggin).

Oh, and yes, I did. Especially the reversal in the final paragraph, that was a neat embellishment. It's a lower case j, though

 

[boooeee]

Yes but that explanation invokes multiple trials, which is not required. And which Prometheus specifically tries to exclude.

Which seems like an arbitrary and pointless requirement. I don't know how you answer a probability question without considering multiple trials.

 

[Squeegee Beckenheim]

I wasn't going to play for a car. Each show up with 100 individual pound notes.

I would posit that if you were to turn up with 100 pound notes which were legal tender, then the time machine you used to get there would be more valuable than any potential monetary win.

 

[jiggeryqua]

I love maths, by the way. I'm not much good at it, just as my musicianship is very poor, but the beauty in both can move me profoundly. Even a good solid subject like physics can generate a community-wide misapprehension based on a seemingly insignificant mistake (OK, I may have got that from Big Bang Theory, but it's possible), whereas maths is dealing with the only real truth we have. Please don't tell me I haven't been keeping up and current thinking is that maths is probably unreal, I couldn't bear it.

I like mathematicians too, in the same way I like musicians. I wouldn't have that Pete Doherty in the house, and I'm given to understand that Beethoven was a bit of an ass-clown socially, but generally speaking, I approve.

I am genuinely interested in how all specialists can improve the communication of their theories or findings or such to the man on the clapham omnibus and soccer moms and Joe Bloggs and Jane Doe. All of whom will benefit most from personalised solutions, or at least a general message crafted by a professional (though one-size rarely does fit all).

I have never suggested anybody's communication sucks. I did chuckle while typing one pertinent remark on an earlier page, but I can't believe that came across in text or merited the word 'sucked' as synonymous with what I did say. I have been straightforward in my assessment of attempts to satisfy my understanding (constructive criticism, to those who volunteered an attempt. I don't give out gold stars for effort though). I have been prompt and sincere in my appreciation of those who were of most benefit, whose work I admired.

It seems to me sufficiently on topic to consider why the MH problem confounds people, and consequently to explore ways of communicating an understanding - most especially so on a discussion forum. Communication is not a separate subject, it's a tool within all of them. Even maths.