Another Monty Hall thread

The Game Room
21

Little Nemo - What exactly are you trying to tell us? That changing the wording slightly or having different assumptions can necessitate an entirely different solution? Yes, of course, 100% agreement. Now where exactly do you desire to go with this? If you’re talking about a literal analogue to the Let’s Make a Deal hosted by Monty Hall, well, we have a problem there as 1. it’s not on anymore, and 2. like any other long-running game show, it had a number of format changes.

Or do you just love abstract theoretical brainstorming? Cool. Fun is fun. Not my thing at all. If that’s what this thread is about, I’ll just quietly excuse myself, but it would’ve been nice to know going in.

Chronos - Geez, is this a No True Scotsman thing now? :weary: The original version I remember seeing was exactly as I described it, and as it’s the simplest one (I also saw it presented that way in an issue of Games magazine, BTW), I’m inclined to hold to it. Of course, that does not in any way preclude the possibility of variants, and in fact for a problem that is so simple I’d expect many of them. Much like I’d expect a substantial majority of knights and knaves puzzles to be trickier than “The stranger mumbles something incoherently. You ask what he said, and he responds ‘I said that I’m a knave’. Is he a knight or a knave?” But the idea that there’s only one Monty Hall or three doors or whatever problem, and the one I presented isn’t it…no. That’s a junk argument, and if I need to accept those terms before entering one of these discussions, forget it.

As for “Cecil Adams”… :angry: Good gods… :man_facepalming: You are aware that we have actual information about the man named Mark Twain, correct? Check it out, yo! You are aware that there’s a difference between going by an invented name (I never once doubted the existence of Pee Wee Herman, Hulk Hogan, Ice-T, Prince Poppycock, or The Micro Machine Man, among many others) and being this eternally mysterious, invisible figure which we know zippo about but nonetheless must treat with the utmost deference? “The smartest man in the world”, when there is literally no evidence whatsoever of a single word of that phrase? And the way he (if “he” is even the right word) turned the whole three-doors problem into a colossal harrumphing ego clash against Marylin Vos Savant killed whatever tiny grain of respect I may have had for him. You are asking me to kiss the boots of a literal invisible space fairy. Screw that forever. You couldn’t pay me to lower myself to that level.

Okay! Lowering the bar! What is the exact wording of the Monty Hall/3 doors problem? Once we get that nailed down, I think this thread will go a lot more smoothly. :grin:

22

No, not at all. I’ll be honest; I don’t see how you got from my OP to where you’re at.

Let me try again:

There’s an abstract probability problem called the Monty Hall problem.

There was an actual television show called Let’s Make a Deal that featured a guy named Monty Hall. There were events on the show that resembled the situation in the Monty Hall problem (hence the name).

I was wondering if anyone had studied the real world show and collected any data on how the statistics of the real world show differed from the probabilities in the ideal problem and what factors may have accounted for those differences.

23

Since everybody gets the details, and history, wrong:

Since no game like your “abstract probability problem” was ever played on “the actual television show,” there are no such statistics. Yes, there was a game that used three doors, but that was about the end of the similarities.

The biggest issue is that it is not legal for a game show host to try to - or even appear to - influence the result of a game based on contestant choices. He can try to influence the contestant into making a bad decision, which Monty Hall did frequently. But he can’t choose an option based on whether the contestant’s first choice was right. This fact actually eliminates all of the objections people have about the rules of the abstract game.

But Monty Hall’s real-life tactics were used correctly when this problem originated (a 1975 letter to the editor of American Statistician). There were three boxes on a tray. One held the keys to a new Lincoln Continental. The contestant chose a box, and Monte [sic] offered to buy it for $100. Then $200, and $500. This was typical - if the contestant wanted to sell, she had to guess how high Monty would go. The change was that in the letter, Monte “did a favor,” opened a losing box, and claimed her chances were now 50%. And he offered $1000 for the chosen box. At this point the contestant - not Monte - suggested the switch.

24

You could certainly do that “Monty Hall Problem” in a real game show. In the problem, Monty always offers the choice whether you pick right or wrong first. There’s no influence here.

Survivor actually did the Monty Hall problem in the last couple seasons. Funnily enough, both times the contestant picked right first and chose not to switch.

25

You play it, if it was clear that the host always opens a losing door that the contestant didn’t choose first. Anything else, and it qualifies as having the production staff (which includes Monty) affecting the contestant’s chances based on what they already did.

My point wasn’t that these clear rules wouldn’t be legal, it’s that anything else would not. So it is safe to assume them in a probability puzzle.

That’s why the real Monty Hall only offered money, or the chance to play a better game.

26

I’m not too sure what your point is. Are you trying to excuse vos Savant from failing to specify a critical element of the problem on the basis that people should be expected to know the legal framework for game shows? That’s… a bit thin.

27

My point is that this is supposed to be a simple probability puzzle, not an exercise in proving how smarter you think you are than Marilyn vos Savant.

That means that if the game can be fair, but only if you make some assumptions that are not contradicted in the problem statement? Go with those assumptions. Especially if there is no real-world experience of the opposite. It means that if specific information is needed if you allow a decision point - like how or when Monty hall decides to open a door or reveal the car - but that information isn’t given? Then it isn’t a decision point at all. Ignore it.

I’m saying that if you compare what you imagine could happen, to what does happen in real world game shows, that what you imagine never happens. Whether or not one realizes that it is because games shows can’t do certain things is not the point. It is me explaining to you why you have never seen it happen, and why reasonable people assume them without thinking.

All of the information you need is in MvS’s statement. If the host could choose to not open a door, that needed to be stated or implied. Was it? No. So he can’t. If he could reveal the car - besides making it a stupid game - that also would have to be stated or implied. Was it? No. So he can’t. But he needs to know where the car is to do all this. Was that stated? Yes. So he can. See how easy this is?

I suggest to you, that by considering all of these alternatives, that you are more of an issue than MvS. Most of them would make it impossible to answer the puzzle. That is being obstructive. It does far more damage to the public’s understanding of probability than MvS ever has (and she has done a lot).

If you want to point out that she makes these assumptions, go ahead. But don’t say she can’t justify them in a simple probability puzzle.

28

What a bizarre rant. The simple fact is that von Savant failed to specify a critical element of the problem, and it would have been much clearer to do so.

No, it is not. It is possible to follow a dubious chain of logic to infer that information, but the information is not there. And since that dubious chain of logic has nothing to do with the essence of the probability problem, it would have been far better to state it explicitly.

29

I roll a fair die. What is the probability that I roll a 1?

The correct answer is not “1/6”. The correct answer is “I don’t know, how many sides does the die have?”. Because anyone who knows me would know that I have dice with many different numbers of sides.

30

This would be interesting if it’s been done. I think it boils down to ‘does the contestant trust that Monty’s intentions would help them’ (i.e. Monty knows they picked the wrong door and offers them a chance to switch). Given that most of the contestants probably don’t understand the statistics, in the long run it would be a study of whether the contestants trust Monty’s intentions.

31

Much like how you fail to specify what that critical element (you used the singular) is, and how it is not covered in what I said. Which may explain why you think it was “bizarre.”

While Monty Hall was not mentioned, and in fact never played a game like this, his methods were always quite clear and consistent. And governed by the guidelines I mentioned.But you did mention several elements before.

He must know what is behind each door. This was explicitly mentioned in the problem statement. Did you read it?

He must be bound to proceed mechanically to always reveal a goat in every iteration. And what suggests to you that he is not? More to the point, can you answer the question if he is not so bound? While you were ignoring the written things you don’t like, did you notice my arguments for why these facts comprise a clear implication that these are, indeed, the rules of the game being played in the question? Moreover, what is the point of mentioning, explicitly, that he knows what is behind the doors, if not to support this implication?

Finally, do you have any indication that anybody who answers “Is it to your advantage to switch your choice?”, regardless of what that answer is, thinks these are not the rules?

If not, you are the one creating red herrings. Hence the “rant.” Get down off of that high horse. You can find similar “failures to specify” in any probability puzzle that is not at least three pages long.

32

JeffJo, I have to admit I’m not seeing what point it is you’re making.

I started this thread (over two years ago) to ask if there had been any comparisons of the differences between the way the abstract Monty Hall mathematical puzzle worked and the way the analogous events that occurred on the show worked in the real world.

You seem to have posted that the events on the show were different than the mathematical problem. That’s correct; I feel that asking about those differences indicates I was aware of their existence.

I feel this is just a rehash of the posts DKW wrote back in 2021.

33

And this was not stated in the framing of the problem, which (if I recall correctly) was in the form of a letter sent in by a reader.

My comment was that the column explaining the probability calculation would have been improved by adding an introductory sentence clearly stating this necessary assumption about the host’s behavior. Your hyperbolic reaction seems disproportionately defensive. Is Marilyn Vos Savant running a cult now?

34

No argument. I even said as much. I’m just saying that it can be accepted as is, but only if you accept that both the problem poser and the problem solver can act reasonably. Especially when poor MvS has limited space for her column.

But I am not defending her - I’m trying to address the actual probability problem, which is being avoided by the (hyperbolic) claims that a reasonable solver can’t infer the intended rules. Which you clearly have done.

Nor am I in a cult of MvS. She makes tons of mistakes, and refuses to acknowledge them. I also said before that she has done lots of damage with these mistakes. Her first answer to the MHP didn’t solve it, it asserted the answer and suggested a different way to look at it intuitively. Her second did at least attempt to solve it, but was technically wrong. The chances she said can’t, actually can change. But there are reasons that are based on other assumptions for why they don’t.

The only issue I have is that these details can’t enter the discussion when one side insists that the game’s rules, which that side acknowledges are the intended game rules, keep getting debated.

The correct answer is that the host could have opened a different door if the contestant initially chose the car, but not if she chose a goat. Assuming (this is the most necessary assumption) that he chooses randomly between the doors he can open, it is twice as likely that she picked a goat than the car. But without that assumption, the chances she picked a goat can be anywhere between 50% (if he always opens that door if possible) and 100% (if he never opens it unless forced).

In fact, the MHP problem belongs to the same family of problems as the Boy or Girl Problem. And she gave a bad answer for it - essentially choosing the solution method that says switching can’t matter. Both depend on whether we must receive a specific piece of information whenever it is true (goat behind #3, there is at least one boy), or if we could, with 50% chance, receive instead an equivalent piece of information if both are true (goat behind #2, there is at least one girl). Martin Gardner actually retracted the “1/3” answer to that problem, and said - as you do, but for better reasons - that the problem statement needed to make the choice of information clearer.

You might want to look up Martin Gardner’s column titled “Probability and Ambiguity.” It is in several of his collections from Scientific American. He explains why the Two Child Problem - which also didn’t explain and “rules” was ambiguous - had the one problem of not telling us how we came to have the information. It then goes on to very clearly express the Two Prisoner’s Problem, with all the details necessary. It is, quite literally, the same problem as the Monty Hall Problem. And it took almost a full page of the book to explain, and Gardner was very good at explaining things concisely.

35

Right, but here we have a case where, even under agreed reasonable assumptions, many smart people are getting it wrong. That’s why I think it was particularly important to spell out the necessary assumptions so that they are front and center in people’s minds, in order for them to think through the implications.

I found that the easiest way to grasp the 1/3-2/3 solution was to realize that if you switch you are effectively getting the best of the other two doors. But it’s easier to “get” that once you realize that the “host” is an automatic procedure that simply tells you which of the other two doors is best (if they are different).

36

When a problem contains incomplete information, the correct response is to state that the information is complete. If one wishes to be thorough, then one can also state possible assumptions that could be made about the incomplete information, and provide a solution for each of those possible assumptions, or to express a range of possible answers to the problem. But that comes after pointing out that the problem as written cannot be solved.

In this case, for instance, one reasonable assumption might be that Hall is forced to always open a door and offer a switch. In that case, the solution is that it is better to switch. Another reasonable assumption might be that Hall only opens the door and offers the switch when doing so would be disadvantageous for the player, because he’s trying to trick the player into switching. In that case, the solution is that it is better to stay. Yet another reasonable assumption might be that Hall is trying to trick the players, but also knows that if he’s as predictable as only offering a choice when it’s against the player’s interest, that he won’t trick many people long-term, and so he only does so when he thinks that the player will fall for it. In this case, we still don’t have any solution, but need to know how good Hall is at assessing players’ reactions.

37

He did sometimes. There is scarcely a scenario that they didn’t try out sometimes.

38

This is another point that I already made. It means that the controversy has nothing to do with any misunderstandings about the rules of the game

That’s part of what I have been trying to say. That most people don’t realize they are taking the rules of the game as implied by the actions in one game. Probably because they are reasonable assumptions for the situation. And still they don’t always solve it correctly. So repeatedly harping on the fact that you do have to make such assumptions only gets in the way of convincing these smart people what the correct solution is.

And this is incorrect probability theory even though it ends up with the right answer. Why doesn’t (to continue with incorrect theory) the probability of the opened door get split between the two others? Why doesn’t it shift to the contestants door?

Probability “combines” that way because another reasonable assumption, that you don’t “spell out”, needs to be made. That the host, when confronted with two doors he can open by these rules, flips a mental coin to choose which one.

ALL OF THE PROBABILITIES “CHANGE” AS A RESULT OF REVEALING WHAT IS BEHIND DOOR #3. (BTW, did you notice that the problem statement doesn’t say the doors have numbers on them? You assumed that.) They change from unconditional probabilities to conditional probabilities. Sometimes the value of these probabilities are the same both ways. They still change.

  1. Door #3 changes from 1/3 to 0.
  2. Door #1 changes from 1/3 to Q/(1+Q), where Q is the probability the host will open door #3 when the car is behind door #1.
  3. Door #2 changes from 1/3 to 1.(1+Q)

By assuming Q=1/2, we get 0, 1/3, and 2/3, respectively. But if Q=1, we get 0, 1/2, 1/2. That is, the probability of door #3 gets split between the other two doors.
+++++

I really don’t understand this. Do you mean “If we take what is explicit here to be the complete information, then the solution is X/Y/Z/Unattainable”? Then in this case, it is “unattainable.” That’s why we can’t do it here.

But I’ll point out that the reason we ask for probabilities, is that some information is not complete.

And that is not legal. The production staff cannot try to affect the chances of winning. That’s favoritism. Once upon a time it was not prohibited, and there were scandals that led to regulation.

39

No. He never did. Check out that video I posted earlier, where Monty Hall himself says that no game like this was ever played on Let’s Make A Deal

40

I didn’t claim that this specific game was ever played.