Since deals are in the news, let's revisit the MONTY HALL PROBLEM...

[brooklynite]

The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from a reader's letter quoted in Marilyn Savant's "Ask Marilyn" column in Parade magazine in 1990.

“Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

https://en.wikipedia.org/wiki/Monty_Hall_problem

[mr_lebowski]

1. Without looking, I know the answer to this ...

Last edited Fri Jan 19, 2018, 06:58 PM - Edit history (2)

And the 'right answer' depends on Monty's motivation(s) (i.e. does he always want you to lose) and whether or not it's entirely his 'choice' to allow you to choose the other door, or if he must do so every time, no matter what.

EDITED to remove some confusing stuff and elaborate:
(As long as he must always conduct the game pursuant to these rules, then the correct answer is ... SWITCH. You're twice as likely to get the car.

If, however, he has the CHOICE of either exposing the first goat and/or letting you choose the other door after showing you the goat ... or NOT ... then his motivations come into play, and the odds (and correct answer) change entirely.

Note that the original problem does NOT state the critical assumption(s) that he MUST have no choice but to conduct the game in this way ... in order for the 33(stay)/66(switch) condition to be 'created'.
)

It is an interesting problem though (esp. for one so basic) and it literally has been debated TO DEATH over the years. Some folks can't wrap their head around the reason why the odds work out the way they do and argue til blue that the 'correct answer' cannot BE ... correct ... even though it is, which is also kinda interesting.

[zipplewrath]

2. You're right

Assumptions are the underpinning to any thought problem.

But when you say "some folks", some pretty educated folks with extensive backgrounds in probability analysis were caught off guard by this one. It is an excellent example though of why reasoning MUST be tested. It's pretty easy to write a program that tests this (including the assumptions you mention).

[mr_lebowski]

4. I know, and they IMHO were 'right' to criticize because there ARE in fact unspoken assumptions ...

Now that I think on it a bit harder I think I misspoke earlier, and the assumption is that Monty MUST always conduct this game in this fashion (you pick, door w/goat is opened, and then he gives you the option to switch doors ... no matter what).

Without the needed but unspoken assumptions, the odds don't change in the way the original writer of the puzzle ... presented it.

Or at least, the ones that figured out that the 'issue' was the unspoken assumptions. Some of them insisted it was wrong even WITH all the assumptions intact. And that subset of folks ... was incorrect.

[jberryhill]

8. His motivation is irrelevant

In the problem, as stated, the host always shows you one door which does not contain a prize after you have made your choice, and offers the choice.

The problem is most easily understood to generalizing to an infinite number of doors.

So, let's play this game:


1. I have a secret number for you to guess between 0 and infinity. Pick one.

2. You pick a number.

3. I now say, okay, the secret number is either (the number you picked) or it is 197. You can keep your guess or switch to 197.

What the infinite-door Monty Hall problem makes clear is that your initial chance was zero. Since all the non-winners except your number or 197 have been eliminated, your chances don't improve by sticking with your original choice.

So, how does this work out with 3 doors? Simple.

2/3 of the time your initial choice is wrong.

That means 2/3 of the time, Monty is constrained to showing you the ONLY other door without a prize behind it (because 2/3 of the time, you picked one which is wrong).

What that further means is that 2/3 of the time, you would do better by switching to the remaining closed door, because 2/3 of the time that is the door with the prize behind it.

I have seen variations of this since the 1980's, have done computer simulations of it, and seen various incorrect explanations of it.

"And the 'right answer' depends on Monty's motivation(s) (i.e. does he always want you to lose) and whether or not it's entirely his 'choice' to allow you to choose the other door, or if he must do so every time, no matter what."

But that is the problem as stated. On each trial, you pick a door, and are shown a door which (a) you did not pick and (b) does not have a prize.

The key to understanding it is that 2/3 of the time, your initial pick is wrong, and thus you are being shown the only other door that does not have a prize behind it.

And, no, the question as stated does not correspond to game play in Let's Make A Deal to begin with. The way that the Big Deal of the Day would work is that two contestants with the highest earnings in the show thus far were each selected to choose one of three doors. If either of the two highest earners declines to participate in the Big Deal of the Day, they kept what they had. After those contestants made their choices, the opening of doors was simply determined by either (a) showing the non-picked door if it was really good, or (b) showing the less valuable winning prize first, depending on which worked better for dramatic effect. Confusing this problem with actual game play in Let's Make A Deal is pointless, since it does not describe an actual scenario of that show.

[mr_lebowski]

11. I'm with ya ... but key fundamental assumption is that ...

Last edited Fri Jan 19, 2018, 06:34 PM - Edit history (1)

He must play this game in this exact fashion, which was not explicitly stated in the original description of the puzzle at hand, yet it HUGELY affects the odds and outcomes.

"“Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?"

Notice that nowhere does it say that Monty MUST offer that choice, nor is it explicitly said 'he always wants you to lose'.

So that's the major source of disconnect ... If Monty IS at liberty to DECIDE to simply to NOT offer you the option to switch, AND it can be assumed 'that he wants you to lose', then it's 100% certain that you should NOT switch, that you instead have a 100% chance of winning by not switching.

Also note we're also leaving aside the question of whether or not he MUST do the initial exercise of opening one door w/a goat behind it first, which is also not addressed by the initial question.

However, since, in 'the problem' as stated, 'he has done so' ... it's probably logical to ignore whether or not THAT action was optional on his part and just assume it was 'required' of him.

But overall if THAT were also optional, and his motivation is always 'that you lose', that would also dictate the odds, because he'd never even DO that part ... if you'd chosen incorrectly the first time, he'd just reveal that you gotten it wrong, and you'd go home with one goat too many.

Thus ... he would ONLY do that initial exercise IIF ... you'd chosen correctly on your first attempt.

That's what I was getting at, sorry for any confusion.

[jberryhill]

12. But even if you don't know

“Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I see what you are saying, but absent knowing his motivation, you still switch, since 2/3 of the time you were initially wrong. That's all you'd really know, absent an explicit statement that this happens every time. I read the problem as directed to the N people reading this thread, so, yes, he does it every time as stated in the relevant context posed to multiple people. But, sure, it does define the problem to make it clear that he does this every time and is required to do this every time.

Anywho... I think I've sworn off "Monty Hall Problem" discussions about five times since, Jesus, 1985 or so when it was a recurring Usenet discussion.

I've developed kind of a weird hobby around the show, which I used to love as a kid. On those cable channels that replay old game shows, I've noticed that in the pre-internet age, it was common to introduce people by their full name and municipality of residence. So, while the program runs, I Google that information to see what is findable about where their lives may have gone since winning that lovely Karastan carpet and running off to Acapulco.

The other thing that's weird is when the prize is something like an AMC Gremlin, my initial reaction is that they were zonked instead of actually winning a valuable new car.

What's disappointing, though, is that when I was a kid and the people won a donkey and a cart, I thought they REALLY got the donkey and the cart. I mean, hey, if the prize they missed was something like luggage, carpet or furniture, then to my mind at the time, getting a donkey and a cart seemed like a much better deal.

I wonder how that sort of thing played with rural audiences? I saw one a few weeks ago, where the prize was a cow and a milking pail. Well, shit, I looked up how much a dairy cow costs, and it seemed like a better prize than a set of Teflon cookware.

And, last question, what IS a "year's supply" of Jolly Time popcorn? Is it all the free popcorn I could possibly consume in a year, or is it some estimate based on how much popcorn the average popcorn eater eats in a year? I figured that they probably gave you a bunch of it, and if you ran out before the end of the year came around, then you'd have to call them to send more.

[mr_lebowski]

16. Good stuff, thanks for sharing.

Last edited Sat Jan 20, 2018, 12:36 AM - Edit history (3)

Think we're saying the same thing ... I 100% agree that his motivations are totally irrelevant if he has NO ability to conduct the game any differently than described in the problem.

Edited this part as I realized that actually there's only 1 mathematically important distinction:
DOES the Host have the option to expose your initial wrong answer ... immediately?

If so, then the Host's motivation (if you can know it) becomes 100% key as to what you should do. IF you've reached the point described in 'the problem', then if he dependably wants you to lose, you have a 100% chance of getting the car by staying pat, because you would not have gotten the host past 'the first decision' above ... had you chosen incorrectly the first time. And if he wants you to win, then you have a 100% chance of winning by SWITCHING for the converse reason ... cause he'd have shown you that you won in the first step if you had.

However, if his motivation is unknowable and/or random like he just flips a coin each day to decide whether he will expose a win/loss at the point of his first option, then you're right back to the 66/33 split. As long as options 2 and 3 'actually happen' it will always be that 66/33 ratio UNLESS the host has a choice in step 1 ... AND you can know his motivation.

So far we've discussed two binary options wherein we're assuming the contestant is AWARE ... which way the 'rules of the game' ... work. But I think where the problem gets interesting is the case where the contestant literally has NO IDEA whether or not the host has the option to immediately expose an initial right/wrong answer. In that instance, does it create a de-facto scenario that's identical to the case where the Host indeed has full control of all his choices ... or not?

Edit: After thinking about that last question further, I think it's still irrelevant what the contestant knows about 'the rules' ... if they do not know the hosts motivation.

I.E. in any case I can come up with EXCEPT the (unlikely) scenario where you both know that Monty prefers that you LOSE, and you also know for certain that he has the option available to him of immediately exposing a right/wrong answer ... then if you found yourself with this option presented to you ... you should be switching.

[jberryhill]

17. But what if the contestant WANTS THE GOAT?

Honestly, more often than not, I thought the zonks were the best prizes on the show.



That's a beautiful goat. Really.

I mean, you have simply not considered the IMPLICIT ASSUMPTION that the contestant DOES NOT WANT THE GOAT!

So, whether or not the contestant knows the rules, you also have to know the contestant's definition of "winning".

For example, I've seen people from Kansas win jet skis for crying out loud. Guy from nowhere near water, might even be a farmer. You think he wants scuba gear or a freaking goat!

[mr_lebowski]

18. Alright, man ... you've convinced me ... we must consider whether the Goat ... is actually WINNING

However I must warn that if we stipulate to THAT ... why, then our odds and probabilities get ALL f***ed up ...

You sure, after all this time and thought you've invested ... that you really wanna re-ponder all the ramifications of this particular wrinkle?

It could get ugly is all I'm sayin' ...

[W_HAMILTON]

14. Thanks for that explanation.

I went to the wiki link and read through it a bit, and it still didn't make much sense to me, but your analogy and subsequent explanations do.

It's still a little weird to me -- I guess that's why they call it a paradox! -- but at least I understand the reasoning now.

[jberryhill]

15. Over the years...

(and the point about the assumption that he does it every time is a good point)

...I've seen this thing so many times, that what I find more interesting are the various ways of explaining that answer (subject to the assumption he does it every time), because people have different approaches to understanding it.

[Farmer-Rick]

3. Why didn't he open the door you asked for?

If he already opened a door, you got a 50/50 chance of getting the car. But it seems you already picked 1 and he opened 3. So, he's not listening to what you say anyway. So, change your answer because he's not opening the door you ask for, so it doesn't matter.

[mr_lebowski]

5. As long as Monty must always conduct the game in this exact fashion ... you should switch.

At least, I THINK that's the key assumption is that he must always 'do it this way', but I might be the opposite I forget.

It's not that 'it doesn't matter', it's that you're 2X more likely to win the car if you switch.

That's what trips people out about the problem.

Funny thing is, this is NOT how Monty ran this game, as the shows producers were not oblivious to these odds. In fact Monty had all options available to him in order to conduct the game in the way most likely to make you LOSE, including simply opening the door you chose right away ... when it was wrong.

[jberryhill]

9. Correct

Not only is it irrelevant to Let's Make A Deal, but as far as the production of that program was concerned, they would prefer that the contestants WIN EVERY TIME.

Why?

Because, except for cash, it's not like the show was buying those prizes.

The entire POINT of the program was to have people giddy with excitement while Johnny read the manufacturers' ad copy for the goods.

You are Samsonite.

You want your luggage advertised.

The reason you are giving the show free luggage as a prize is so that Wilma from Fremont, California will jump up and down in a rabbit costume while Johnny says, "It's SAMSONITE luggage! The toughest safest luggage there is. If you have places to go, then you want to take SAMSONITE luggage with you."

What you don't want, if you are Samsonite, is for Wilma to get a goat, and then for Johnny to say, "Oh, by the way, there was some luggage in the box where Carol Merrill is standing."


The POINT OF THE SHOW was to ADVERTISE THE MERCHANDISE. That works best when people are winning it. There wasn't some executive at Goodman Productions saying, "Hey, Monty, go easy on giving out the cars today."


But, again, this problem has nothing to do with the television show or the way it was conducted.

[longship]

6. You double your chance by switching doors.

This is simple.

At the beginning you have a 1/3 chance of selecting the 🚪 with the 🚗 and a 2/3 chance of selecting a 🚪 with a 🐐. Now Monty knows the 🚪 where the 🚗 is so no matter which 🚪 you select he can always open a 🚪 with a 🐐. In the 1/3 times you pick the 🚪 with the 🚗, switching will get you the other 🐐. However, for the 2/3 times you pick a 🚪 with one of the 🐐, Monty has to reveal the only other 🐐and switching earns you the 🚗.

Therefore, you double your chances by always switching. 2/3 times you will get the 🚗. The only way you lose is in the 1/3 original chance where you first selected the 🚪 with the 🚗.

[Leith]

7. The Mythbusters Did This Experiment

spoiler: switching is better.

https://mythresults.com/wheel-of-mythfortune

You can also put "Monty Hall Problem" into YouTube's search engine.

[Adrahil]

10. Longship is right.

Switch. You double your chances by switching. The key to this problem is that Monty has privileged knowledge and always reveal a door with a goat behind it. 1/3 of the time you will pick the correct door at the beginning, which means a 2/3 chance that the car is behind one of the other two doors. That means 2/3rd of the time, if you switch, you will get the car.

CLassic probability problem.

[jberryhill]

13. What if you want to win the goat?

If you have a goat, it will keep your lawn trimmed, fertilize it, dispose of leftover food, and provide milk for cheese and eventually meat for curry and hide for leather.

I'd ask the host if I could just have the goat.

The car is going to require registration, insurance and title fees, and I already have a car. I don't have a goat, though.