Another Monty Hall thread

The Game Room
41

So what specific game do you think you recall?

My point is that he never let you switch your choice within a specific game. He would offer you money if you gave up your (as yet unknown) prize. He sometimes even eliminated other possibilities, to tempt you more. He let you trade in your (known) prize for a chance at “The big deal of the day,” which is where the three doors come from. But two players were chosen for that game, not one. He did reveal doors, but if I recall properly it was in increasing value. The first might depend on who picked what, but the Big Deal was always revealed last.

What I am trying to get across, is that none of the odds-altering revelations that people get so worried about ever happened. No switching, ever. And if you think you remember them, you are altering your recollections to fit what you suspect could have happened.

42

It is silly to call it “incorrect probability theory” to realize that what the host offers you in the switch is the best of the other two doors - i.e. success if the prize is behind either of them. And to realize immediately without convoluted algebra that this implies a 2/3 probability of success.

Because that would be incorrect probability theory.

It’s odd that your “correct probability theory” leads you to this conclusion, since it is wrong. The probability of the door you originally selected does not change - it remains at 1/3.

That’s because the new information that the host has given you by opening a door is only about the OTHER two doors - which of those two doors is best (if they are different).

43

This again? It is preposterous to suggest that a key piece of information about a toy probability problem need not be stated because readers should know the arcane details of statutes that govern what actions are legally permitted in American game shows.

44

The production staff is doing no such thing. It is illegal to circumvent the regulations of the broadcast standards and practices - but performing the Monte Hall problem doesn’t do that, because the process of the MHP is a known process to the regulators. It would be illegal if they changed the process without informing the proper regulators.

45

It is possible, using just the given information, that the probabilities are not {1/3,2/3}. The conclusion you reach is wrong. Since you are convinced that {1/3.2/3} is correct, you are using incorrect probability theory. You can - with an additional assumption - show that under reasonable interpretation that {1/3,2/3} does turn out to be correct. But only if you understand and apply correct probability theory.

Now, (1) you are convinced that {1/3,2/3} is correct, and (2) you seem to get very upset when it is pointed out that what you have long accepted as true is not. These facts make for a clear demonstration of why you need to use correct theory even when it doesn’t appear necessary.

But it really isn’t all that convoluted. If the contestant originally picks incorrectly, the host must open a specific door. That’s a weight of “1.” If the contestant originally picks correctly, the host has 2 choices. The relative chances, between these two remaining possibilities, depend on how that choice was made. NOT on the just the original probability of one of them.

Assuming he picks randomly between these two - that is, each has a 1/2 chance - that possibility has a weight of “1/2”. It makes the odds 1:1/2 (or 2:1) for the first possibility, that the contestant picked incorrectly. But if he does not choose randomly, these odds are different. If what turns out to be his choice had a probability of Q, the odds are 1:Q.

. Both are a result of applying incorrect theory. One gets an answer that can be proven with correct theory, but you don’t think that is necessary. You just assert that one is right. The fact that gets the right number does not justify the assertion.

Again, . It changes to Q/(1+Q), where Q is the probability that (using the doors in the problem statement) the host opens door #3 after the contestant chooses door #1, and it has the car.

Try it. Get three cards, two 2s and an Ace. Shuffle them, and lay them out right-to-left. Place a weight of the left-most card (door #1), and look at the other two. If the right-most card (door #3) is a 2, turn it over. If it is the Ace, turn over the middle card (door #2), which will be a 2. Repeat this N times,

  1. Each position gets the Ace (about) N/3 times.
  2. The right-most card is turned over about 2N/3 times.
  3. The middle card is turned over about N/3 times.

The probability, that the left-most card has the Ace given that the right-most card was turned over, is (N/3)/(2N/3) = 1/2.

Now, the fact that the right-most card is turned over twice as often as the middle card means the average probability is 1/3. But that requires the assumption that the contestant does not see which door is opened. Are you suggesting that is reasonable?

The point is that Q can be any number between 0 and 1. The probability that the car is behind door #1, after the contestant chooses it and the host reveals door #3, can be anything between 0 (if Q=0) and 1/2 (if Q=1). But we aren’t told, explicitly in the problem statement, what Q is. Even though it is necessary to determine the probability that the car is behind door #1, now that door #3 has been revealed. So we make the reasonable assumption (remember that it is good form to acknowledge when we make assumptions, and why?) that Q=1/2, and get that the probability for door #1 is 1/3.

The value is the same, but the reasons why that value applies have changed.

And I’m not suggesting that readers should know the arcane regulations of game shows. I’m saying that in their actual experience, they have observed the results of those regulations, and only those results. “Remembering” instances where those regulations were not followed is selectively changing your memory. It hasn’t happened, because it could not.

What I am suggesting, is that the people who arrive at an actual answer - right or wrong - are all making the same assumptions about the rules of the game. Usually by tacit assumption. So the unstated rules of the game are not the cause of the controversy.

What I am suggesting, is that raising the issue of the lack of an explicit statement of the rules does nothing to resolve the controversy about the problem. If you feel it is important to point them out, even if your audience has already made them, go ahead. Just don’t blame the problem statement for how you recognized a detail that has nothing to do with the controversy.

46

Oh good grief, I’m obviously now talking about the situation where we DO know that the host mechanically reveals a goat every time, and where the correct answer is well known to be 1/3 - 2/3.

It’s very difficult trying to communicate usefully with someone who responds every time with a long, confusing and patronizing essay. There is no unique way to do “correct probability theory”. The math here is trivial, and using confusing algebra and over-generalization does not grant insight. What is required is clear knowledge of the exact behavior of the host and the information that derives from it.

47

It’s a counterintuitive statistics thought exercise that ONLY happens under the postulation of a bad door always being opened. Changing or omitting any of that information makes it a different equation.

You should zombify a thread about the airplane on a treadmill, or jump into the alternate ending of Big debate.

48

And alternative answer possibilities apply even when “the host mechanically reveals a goat every time.” Despite it being widely accepted that “the correct answer is … 1/3 - 2/3,” it doesn’t have to be true. I’ve tried to explain why, but you won’t consider it. This isn’t my original idea: try looking up this paper: https://www.tandfonline.com/doi/abs/10.1080/00031305.1991.10475821

It’s just as difficult to communicate with someone who thinks they know an answer, and refuses to consider any alternatives, or dwells on the inconsequential. We end up combining various issues into one. So I’ll separate them:

  1. What is inconsequential here, is obsessing on the rules of the game. Those who do cannot actually provide an answer because the ambiguity they create does not allow one. But discussing it to death gets in the way of evaluating the various solutions proposed by those who accept the implied rules. Is this short enough for you? It is all I intended to get into when I started.

  2. The reason there are competing solutions is because people try to solve it as an unconditional probability problem. Conditional probability is not intuitive, so they try to avoid it. The way you do so, is to avoid the fact that the contestant sees the door that gets opened. This should eliminate half of all possibilities right there, even before we can apply what the information means. You consider them all, by “combining doors.” We know that this is a highly unsuccessful approach, even if you find it logical, because it only convinces the people who already agree with you. Is this short enough for point #2, which is that there are issues if we don’t use conditional probability?

  3. There is a better way. We first eliminate the cases where door #2 would have been opened. Call this group A, and it represents 1/2 of all cases. Then we note that all of the cases where the car is behind door #2 remain. But half of the cases where it is behind door #1 were in group A, are were eliminated. So the cases that remain where the car is behind door #1 include half as many as where it is behind #2. Hence switching wins twice as often. This feels just as intuitive, and it is theoretically correct. But it does require an assumption (one you ignore in point #1) about how the host chooses.

  4. Most of the “convoluted algebra” I’ve included is trying to point out to you why your solution is incorrect, and teaches incorrect procedures. I started out simple, and it grew progressively harder and longer as you refused to accept it. But it is clear you won’t try, so I give up.

I’ll end by suggesting a book by Jason Rosenhouse called “The Monty Hall Problem.” I don’t know if it was ever published, but I found it online.

49

Buddy, you’re the one who re-opened a discussion from over a year ago with a word salad that I’m still not sure has actually made a point.

50

But some incomplete information can be quantified, while others can’t. Again, if I ask “If I roll a die, what is the probability that I will get a 1?”, we can’t answer without knowing how many sides the die has, or at least without knowing a distribution of possible numbers of sides.

And is it reasonable to expect respondants to a puzzle to know the details of game show law? Or to assume that the puzzle takes place after such a law, in that law’s jurisdiction?

You just said that he wasn’t allowed to do this.