Please help Re: Simplifying an explanation to the Monty Hall Puzzle

[kwbMitel]

I am tutoring a 12 year old that is struggling in math.

One of my first memories was finding my love for math puzzles at that age (or really a bit younger)

The Monty Hall Puzzle is my favourite for showing the counter-intuitive nature of math at times.

I wont be discussing My favourite puzzle that involves alternate solutions. If you know me, you know what that is.

I know starting with the Monty Hall Puzzle is not the ideal choice considering there are bone fide geniuses that don't agree or at least didn't agree at some point. I'm not saying I include myself as one of those geniuses but I too was amoung the nay-sayers for a short time. (Coding to the rescue)

Nevertheless here we are.

Sometimes looking at a thing from a different direction can illuminate a discrepancy (or a solution). There was a lottery odds question at some time where someone asked if there was a different probability of buying half the tickets for one draw or half the the quantity of tickets 1 ticket at a time over X draws.

Long story short, I proposed that there was little to no difference while most were saying that the odds of one at a time never changed from draw to draw for the multi-draw method. The answer was well north of 40% though, not the 50% I claimed and definitely not the >1% the others were saying, but eventually started saying Greater than 40% but less than 50%. The kicker for me was to flip the question and calculate the odds of losing and I got a number that matched the 40%+ crowd. Wins all around.

Flipping the Monty Hall question looks like this.

What if right at the beginning I offered you 2 choices instead of one.

Does that change anything?

One door can still be revealed. It will naturally be one of your 2 doors and we can ask the question again. Switch or No.

I see it as strongly emphasizing the first choice being the important one with respect to odds but I know the answer and may just be fooling myself.

 

[Chris Miller]

Well, in the original probleem description it's often forgotten to tell that Monty knows which 2 doors have a goat and which has the price. And, well, in the classical case when you only open choose 1 door, Monty always opens one of the two remaining doors, never your chosen door.

In case you choose 2 doors, there would only be one door he would show by the classic rule, but you say "It will naturally be one of your 2 doors", so he can open any door, I assume.

Well, and the next definition missing is what is your price in the end? Both doors you choose? Well, then after he shows you a goat door, no matter if that's one of yours or the one you didn't pick first, wouldn't you simply choose the other two doors and know you have the price and a goat? It's really much more trivial that way. And in that case, your final choice of two doors should always be the two doors Monty does not open. That would be independent of your first choice, then, because it only depends on what goat door Monty opens, and that's also not forced by your first choice.

So, I don't know, maybe you should define the rules of your "pick 2 doors" variation more precisely, so it becomes not as trivial.

Chriss

 

[kwbMitel]

Fair Enough,

"It will naturally be one of your 2 doors".

If it is not one of your two doors, and the door revealed is the Car or The goat, There no longer is an apparent even choice between the remaining 2 unknown doors. You at that point know the car is in one of those (your) doors or exactly where the car is.

So the reversal MUST reveal one of your 2 doors or the puzzle ends there.

On the second choice, when there are 2 doors remaining, you are again offered to pick which one

It is never a rule of the original puzzle that Monty cannot open your door but that is implied by stating he always open a different door. I'm asking if that is enough of a difference to confuse someone or that it changes things in such a way that makes it invalid, or if it is not an improvement at all and we are left with the same result.

Just by you asking the question makes me think I already got my answer.

 

[kwbMitel]

Follow-up

Can anyone think of a puzzle, preferably without tricks, that can help someone with some fundamentals such as converting from percentage to fractions to decimals in every direction?

Not word puzzles, he gets enough of those already. Something that requires a greater understanding of the principles.

He's really struggling with this and I want to give him something that has the enjoyment of discovery.

 

[Chris Miller]

I may think about one problem like that.

It just reminds me how you can see why switching is a good strategy in the Monty Hall problem if you blow up the situation to 100 doors from which you choose one. You then clearly have a 1% chance with your initial choice. In that variation, Monty opens 98 goat doors - and does not reveal what you picked, of course. And that, by the way, clearly is part of the classic rules, as the question is whether you stick or switch, why should he tell you what you picked? This would make the question obsolete in the sense that if he shows you the door you pick has the price you, of course, stick to it, and if he shows your door has a goat, you of course would switch. So definitely Monty will never tell you what you picked initially.

Well, to get back to the variation: You picked one of 100 doors. And it still only has a 1% chance to be the right one. The 98 opened doors have a 0% chance to be the right one, what chance does the other closed door have now?

Chriss

 

[kwbMitel]

Ive seen that description fail numerous times. People focus on what remains not what came before. If you walk them through one step at a time, they will always switch up to the last one where they generally stick as they feel it has reached 50 50

I cant discuss all the variations of explanations. They all appear to fail and in my mind, it’s not the quality of the illustration but the willingness to take the counter intuitive leap.

This led me to, if the answer is counter intuitive as stated, can flipping the script then make it intutive yowards the right answer?

No, I dont think it does. The issue remains that when 2 doors remsin, too many people will maintain that it is 50%.

Not everyone has cofing skills or logic analysis as I do. Simply coding the problem made me see the light. The results of that were secondary confirmation

 

[karluk]

I hope I'm not embarrassing myself by giving a snap answer without thinking things through, but it seems obvious to me that if you get to pick two doors, you have a 2/3 chance of having picked the prize. Revealing an empty door doesn't change this, so the contestant should stick with his original pick, and expect a 2/3 chance of winning.

The reasoning looks the same as the original Monty Hall game, the only difference being that the contestant is already holding the door with the 2/3 chance.

 

[Chris Miller]

Well, karluk,

but in the case kwbmitel allows - "It will naturally be one of your 2 doors" - if Monty reveals that one of your doors is a goat. Switching from that door and keeping your other is a natural to cover the case your other door also is a goat. In the end, you always pick the two doors Monty doesn't show and have 100% knowledge what you get - the prize plus a goat. If you stick, there is a chance the other door you picked and Monty didn't reveal also is a goat. Why would you stick to your 2/3 chance when you can leap to a known result?

@kwbmite, I agree with the description of the wrong reasoning people stick to - "they generally stick as they feel it has reached 50 50".
The leap to make to understand this is that the change of information you have assigns new probabilities. The probability of 1% for your first choice of 1 in 100 doors does not change after revealing 98 doors, but the chance of the other remaining closed door does in fact change. It now is almost certainly the prize (Sorry for having said price all the time).



Chriss

 

[kwbMitel]

@ Karluk, If I am reading you right, it looks like your self admitted snap answer, suggests an improvement towards understanding the probability with my new description.

@Chris Miller, anyone who knows the answer will follow your 1-100 explanation and agree with it. You cannot be certain by any means that the logic will be found by a non-believer.

I'll give you 2 sides of that same coin.

You make the statement that "You then clearly have a 1% chance with your initial choice"

You also dispute my statement "It will naturally be one of your 2 doors" or at least say it needs more clarity.

I will state with 100% certainty, that unless you believe both those statements on the face of it, you are not going to be convinced from one side to the other.

So here is what I have now come up with that incorporates what I learned from coding and the 2 statements above that I say must be true.

[ul]
[li]Start program[/li]
[li]contestant makes first choice (random generator)[/li]
[li]You pause the game and offer your nay-sayer the option to switch before the reveal[/li]
[li]If the logic of switching at this point eludes them, you pull out the 1-100 description[/li]
[li]If the logic still eludes them, goto end program else continue i.e. if they wont switch at 1-100, accept that you wont convince them, move on[/li]
[li]The contestant has switched at this point if the program is still running[/li]
[li]Instead of Monty picking a door to reveal, you allow your contestant to know all doors (as Monty does) and ask them to tell you which door would have been revealed? this is my aha moment because as a coder, you must know and allow for all possible outcomes[/li]
[li]The door to be revealed will "Naturally be one of your 2 doors" no exceptions[/li]
[li]It may need a couple of repetitions to get the point across but you are now well ahead[/li]
[li]Print results[/li]
[li]end program[/li]
[/ul]

That's as close as I think I will ever be.

 

[Chris Miller]

Okay, kwbMitel.

The door to be revealed will "Naturally be one of your 2 doors" no exceptions

"no exceptions" makes me think, that you meant something else than I interpreted into "will naturally be one of your 2 doors". I interpret this as it will often be that case, but not always. Not that it has no exception. Revealing the third door would not be done by Monty, Monty then would (naturally) reveal one of your picked goats. But if the third door has a goat, he could also reveal that.

No thinking of that situation it makes no sense for Monty to reveal the third door, if it's a goat, as you then know your doors are a goat and the prize.

Anyway, now that you made clear (but did you actually) the door Monty reveals will always be one of yours, that makes things still as easy as I ended up with: As Monty never reveals the prize, he reveals one door you picked, that is a goat. As you pick 2 doors you always have at least 1 goat door and he can pick it. But then you always don't stick to that door, why would you stick to a goat? You still end up by changing your second door to the one you didn't have in your first pick.


Well, and overall, about the understnading of the "naturally phrase" in your outset:

at least say it needs more clarity.

That's what I said, I never disputed what you said, What you say is your decision and is the rules. I just said that I wouldn't "scale up" the ruleset this way, as Monty does never reveal your door in the original outset of the problem.

Now I believe you want to state this Mont explicitly has to be reveal one of your doors and explicitly not the third. But I'm stll not sure that's what yyou mean with "It will naturally be one of your 2 doors". So it's about the clarity of this.

What remins is, if you know a goat door, and you do after Monty reveals one. You'd be dumb if you don't remove that goat door from your initial pick. Or - in other words - no matter what you initially picked, your choice for two doors after you know one goat is the other two doors. Completely independent from your first picked pair of doors. That remains as is.

Chriss

 

[kwbMitel]

I interpret this as it will often be that case, but not always. Not that it has no exception.

No, it is absolutely 100% fact.

Let's take a step back.

Do you agree or disagree that Monty will NEVER reveal your first picked door.

Presuming agreement: Do you agree or disagree that the remaining 2 doors must contain at least 1 booby prize (goat)

Following agreement: It can be said that if you are allowed to switch, prior to the reveal, that the reveal MUST be one of the remaining 2 doors that were not your first choice and that one of those 2 doors must always be the one that is revealed. "Natural Conclusion, Naturally the case"

And Finally it follows: If you do not switch before the reveal, then you do not understand probability at all and the conversation ends as a pointless discussion. If you switch you now have the remaining doors that were not your first pick. Once the goat is revealed from one of those 2 doors that you now have, you are now in the high probability zone and even if you think it is now 50% you have inadvertently improved your odds of winning. If you now see that it is only your first choice that matters with respect to winning the grand prize, all the better.

And really, if you do not agree with any of those, you need to reassess your understanding of the problem. Withdrawn as unnecessary and inflammatory but left in to frame possible unnecessary responses from Chris, and to retain my responsibility to them

EDIT*

And maybe this in fact is the way to explain it, no assumptions, just 2 solid statements and the conclusion that can be drawn from them.

 

[Chris Miller]

kwbMitel,

Do you agree or disagree that Monty will NEVER reveal your first picked door.

What case do you talk about? The normal one-door picking case, right? Because you talk of a single door, not plural doors.

Then Monty never reveals your door, yes.

I am critizing the precision and clarity of your statement "will naturally be one of your 2 doors" in your two door picking case. But you avoid getting exact on that one. You make up the rules for that and can make them up as you want, but don't use a suggestive phrase in it lke "naturally", just be explicitly clear. That's all I'm asking of you.

If you want to say that in your two door pipcking case Monty will reveal one of your doors, just say "Monty then revelas one of the doors you picked." And that's describing what you want to say precisely.

Chriss

 

[kwbMitel]

Chris, you can't agree with 1 thing I said and ignore the other.

I said that there are 2 absolutely firm statements that can be made about the original problem

1) Monty will never reveal your door.
2) That there is always a door that remains that can be revealed without revealing the grand prize.

If you can agree with both of those statements, and not come to the "Natural Conclusion" that they absolutely dictate, then I'm sorry, I see no more point in debate.

If you do see the natural conclusion that they dictate. Why are you debating at all?

 

[Mike Lewis]

I would like suggest an alternative approach to Monty Hall. The original puzzle asks for a strategy that would maximise the chances of winning a car and minimise that of winning a goat. That seems to me to be the wrong way round.

If you win a goat, you can tether in your garden where it will eat the grass and save you the considerable time and effort of mowing your lawn. The goat will convert the grass into milk, which is a healthy drink, and can also be used to make a delicious range of cheeses, not to mention yoghurt and other dairy products.

If you win a car, you will have to spend money to tax and insure it, and then more money for fuel every time you use. Depending where you live, you might also need to pay for parking and/or road tolls. Then there are ongoing costs for maintenance and repairs - and probably other things that I don't know about.

Give me a goat any day.

Mike



__________________________________
Mike Lewis (Edinburgh, Scotland)

Visual FoxPro articles, tips and downloads

 

[kwbMitel]

Mike, as flippant as your answer is, which may or may not have formed a smirk, it does not in any way get me closer to my stated goals.

Let's restate my latest as follows:

1) Monty will never reveal your door regardless of what it contains.
2) There is always a door that remains that can be revealed without revealing the grand prize.
3) There is a door that contains a prize that has greater value than all others and winning it is the goal.

 

[Chris Miller]

I fully agree with Mike,

and kwbMitel, you're moving the target when its hit. I won't follow you onto your new battleground.
I stick to my demand: State the outset of your Monty Hall problem variation precisely, and we can talk about it again. You haven't done that yet and unless you do so we can talk for years without am outcome.

Chriss

 

[strongm]

>It is never a rule of the original puzzle that Monty cannot open your door

Actually, Steve Selvin (earliest known originator of the problem) and Marilyn vos Savant (who popularised it through her eponymous "Ask Marilyn" column) are very clear that the host opens one of the remaining doors (or boxes, in Selvin's case)

 

[kwbMitel]

Chris,

I suggest that I have not moved the goal posts at all. I have only been following your instruction/demand that I better define what I meant.

If you cannot see the relationship between what I said and what I am saying and how they are simply viewing the same thing from opposite directions then I absolutely agree that debate has reached it's "natural conclusion"

If you also have not surmised that I am not strictly stating that my first offered explanation is by any means good. I offered it for judgement and ultimately concluded that it was not an improvement all on my own. I have said so in fact.

If you missed the reply to you that asked you to take a step back to the original problem and reanalyze from that stand point then, ok, but how is that my problem?

All that being said, you have not in any way provided your answers what I consider 2 simple and reasonable questions.

Do you agree with both of the following:

1) Monty will never reveal your door regardless of what it contains.
2) There is always a door that remains that can be revealed without revealing the grand prize.

Answer or not, don't really care, but like you I will draw my line in the sand. You must either agree, or disagree to continue fruitful debate with me.

I am looking for common ground where interpretation is not required. What are you looking for?

 

[Chris Miller]

I know your overall goal is to make the original Monty Hall problem understandable by learning from another outset, just like the variation I talked about with 100 doors.
I also know you would like to see a puzzle for your son "that can help someone with some fundamentals such as converting from percentage to fractions to decimals in every direction", I haven't lost that from my view just because we continue to discuss the Monty Hall problem here. "Not word puzzles, he gets enough of those already", means clearly not a word problem.

But as we're on the word "naturally" here in our discussion, let me just tell this bout mathematical problems descriptions:
There's never the need to emphasize something is "naturally" that way, that's always a hint on something incomplete in the problem description, as you in short say "I don't have to explicitly say that, it's obvoious". Well, if something is obvious, then you can also leave it off overall. But in problem descriptions that should define precisely how they should be understood or in this case what is done by contestant and Monty Hall, statements that are obvious can be made and will usually be made, even though you may argue they are obvious, just to make the description precise. As I said initially often the problem is nont described in all details. It may be natural to assume that Monty never reveals the big prize, that's "natural". In maths problems nothing is natural though, until it's explicitly stated as the outset of the problem. More important to understand the problem is to know that Monty is in fact knowing what's behind all doors. That really isn't a natural, if it's never stated. So, perhaps you now see why I react "allergic" to your usage of that word. If "naturally clear" statements are made, that's not a sign of disrespect to the solver of the problem and his common sense, it's just precision in the problem description.

Chriss

 

[Chris Miller]

...asked you to take a step back to the original problem and reanalyze from that stand point...

Let me look where you said that...sorry, I don't find that.

For the original Monty Hall problem the two statements

1) Monty will never reveal your door regardless of what it contains.
2) There is always a door that remains that can be revealed without revealing the grand prize.

are both true.

1) is true as it is an outset that belongs into the problem description, or the problem isn't described precisely.
2) is a corollary of 1) and the knowledge there are two doors with a goat and one with a booby prize (goat) and 1 door with a great prize (car): As you can't pick both goats Monty always can reveal a door with a goat among the two you didn't pick. And that's also only easy to see if the problem description tells you that Monty has the knowledge of what's behind each door.

It's all quite "natural" as anybody knows the show and knows how it works. That's why it's picked as the analogy to not need to explain too much, but that's the error, as the actual Monty Hall show final also worked different in some aspects and that's also a typical strain of endless but pointless discussions, if you don't state the math(s) problem as derived from Monty Hall but as precisely as necessary to understand how the two step process works for your "thought experiment".

Chriss