Talk:Monty Hall problem

Latex

Ew looks ugly someone should change the formating to use LaTeX — Preceding unsigned comment added by 2604:3D09:1580:9600:ECAA:144E:36B1:E1C8 (talk) 16:33, 10 April 2022 (UTC)

I think that's wise. In this era of Covid one should always wear gloves. EEng 16:24, 10 July 2022 (UTC)

"Monty Fall" host behavior variant

Firstly, there are 3 sources given, but [53] doesn't mention this variant at all (it also doesn't mention the 2nd variant in the table, "The host always reveals a goat and always offers a switch. If he has a choice, he chooses the leftmost goat with probability p"). As such, I've removed this source from the two variants. Also, [62] simply asserts that the probability for this variant is 1/2 in its introduction, but never explains why. However I've left it in place for now.

[34] does explain the variant via the "Proportionality Principle", but I believe there is a hidden assumption being made, namely that the contestant doesn't know ahead of time that Monty will always fall on a goat door. If the contestant has the same information as the reader and knows Monty is guaranteed (e.g., via divine intervention) to fall on a goat door that wasn't initially chosen, then the variant should be identical to the original problem, only instead of Monty choosing which door to open, it's a divine being forcing him to fall onto a certain door.

As stated, the variant is confusing because Monty cannot both fall randomly (ie. have equal probability of falling on any of the 3 doors), and be guaranteed to always fall on a non-initial goat door. It seems artificially constructed specifically to prove the point that modifying assumptions about host behavior also modifies outcome probabilities. However, the other 10 variants in the table already do this, and have the added benefit of being realistically testable. I am unsure whether this particular variant actually helps in understanding, and am considering removing it from the table. However, I would like to get some other opinions first 30103db (talk) 15:46, 19 July 2022 (UTC)

Update to direct calculation of Monty Hall problem

EDIT: I have just updated the proof to no longer rely on mathematical prose, to include a citation, to improve the organizational structure, and to elaborate on issues that might be unclear to the reader (e.g., why P(Hi|Xj) = 1/2). These improvements address many of the issues raised in the comments.

Hi there. I originally posted updates to a proof for the direct calculation section of the Monty Hall problem page, but I accidentally typed "Monty Python problem", leading a user to revert my changes under the assumption they were not serious. I learned this on the talk page of that user, where others said I was fine to reimplement my changes so long as I fixed my typo. But when I did so, my changes were again reverted because people assumed I was just ignoring suggestions since I did not post on this page. As a result, I am being made to post here to ensure the consensus agrees that my changes are fine before reimplementing them. This is a bit of a headache since this originates from a miscommunication rather than an actual problem with my proof, but that is alright.

My changes accomplish three things. They:

• Improve the intuitiveness of the proof by deriving solutions from claims that more closely correspond with the Monty Hall problem rules
• Prove the probability for all three doors by deriving a probability equation that applies for all three doors, rather than just the second
• Remove redundant text and improve clarity by making explanations more concise (also adding in a section on conditional probability rules)

Here are my changes:

---

Direct calculation

In this proof, we will directly compute the probability of the car being behind each door using Bayesian statistics.

Let us label each of our doors as Door 1, Door 2, and Door 3. Next, let us define three functions. Xi denotes that you initially choose Door i, where i represents any door from 1 to 3. Similarly, Ci denotes the car is behind Door i. And Hi denotes the host opens Door i.

Let us define j and k as numbers, like i, from 1 to 3. i, j, and k will represent different numbers, meaning, for example, that Xi and Cj will represent that you chose Door i which is not the door with the car, Door j.

With these rules in mind, we can define our probabilities as the following:

${\begin{aligned}P(Ci)&={\tfrac {1}{3}}\\P(Hi|Ci)&=0\\P(Hi|Xj)&={\tfrac {1}{2}}\\P(Hi|Cj,Xj)&={\tfrac {1}{2}}\\P(Hi|Cj,Xk)&=1\\P(Ci)P(Xi)&=P(Ci,Xi)\end{aligned}}$

• P(Ci) = 1/3 represents the probability that the car is behind one of the three doors.
• P(Hi|Ci) = 0 represents the impossibility of the host opening the door with the car.
• P(Hi|Xj) = 1/2 represents how there are two remaining doors for the host to choose from once you have chosen a door. It is important to note this expression does not care which door the car is behind, so we do not have to worry about how the possible doors containing the car affect the host's door choice.
• P(Hi|Cj,Xj) = 1/2 represents how, given you chose the door with the car behind it, there are two remaining doors the host can open.
• P(Hi|Cj,Xk) = 1 represents how, given you chose a door without the car behind it, there is one remaining door the host can open.
• P(Ci)P(Xi) = P(Ci,Xi) since our rules do not specify there being conditional dependencies between P(Ci) and P(Xi), meaning that Ci and Xi are independent events.

From Bayes' theorem, we know that P(A,B) = P(A|B)P(B). We can extend this logic to three events using the chain rule: P(A,B,C) = P(A|B)P(B|C)P(C).

The conditional probability, we know that P(A|B) = P(A,B)/P(B). We can extend this logic to three events using the chain rule: P(A|B,C) = P(A,B,C)/P(B,C).

Next, let us derive a formula to calculate the probability that a given door contains a car just in case X1 and H3:

${\begin{aligned}P(Ci|X1,H3)={\frac {P(Ci,X1,H3)}{P(X1,H3)}}={\frac {P(H3|Ci,X1)P(Ci,X1)}{P(H3|X1)P(X1)}}={\frac {P(H3|Ci,X1)P(Ci)P(X1)}{P(H3|X1)P(X1)}}={\frac {P(H3|Ci,X1)P(Ci)}{P(H3|X1)}}={\frac {P(H3|Ci,X1){\tfrac {1}{3}}}{\tfrac {1}{2}}}={\tfrac {2}{3}}P(H3|Ci,X1)\end{aligned}}$

We have done the bulk of the work, proving P(Ci|X1,H3) = 2/3 P(H3|Ci,X1). So the probability of the car being behind a door given you chose Door 1 and the host chooses Door 3 equals 2/3 * P(H3|Ci,X1). We are in luck, since we know the probabilities of P(H3|Ci,X1) for each possible Ci! All that remains is for us to substitute Ci with each of the three possible car-behind-door events and plug numbers into our equations.

To calculate P(C1|X1,H3) (car behind Door 1):

${\begin{aligned}{\tfrac {2}{3}}P(H3|C1,X1)&={\tfrac {2}{3}}{\tfrac {1}{2}}={\tfrac {2}{6}}={\tfrac {1}{3}}\end{aligned}}$

For P(C2|X1,H3) (car behind Door 2):

${\begin{aligned}{\tfrac {2}{3}}P(H3|C2,X1)={\tfrac {2}{3}}1={\tfrac {2}{3}}\end{aligned}}$

For P(C3|X1,H3) (car behind Door 3):

${\begin{aligned}{\tfrac {2}{3}}P(H3|C3,X1)={\tfrac {2}{3}}{\frac {P(H3,C3,X1)}{P(C3,X1)}}={\tfrac {2}{3}}{\frac {P(H3|C3)P(C3|X1)P(X1)}{P(C3,X1)}}={\tfrac {2}{3}}{\frac {0P(C3|X1)P(X1)}{P(C3,X1)}}=0\end{aligned}}$

We have now shown that, given you initially chose the first door and the third door is opened by the host, there is a 1/3 chance of the first door containing the car, a 2/3 chance of the second door containing the car, and a zero chance of the third door containing the car, thereby solving the Monty Hall problem using Bayesian statistics.

---

With that in mind, I would greatly appreciate if people could reply to this post telling me whether they think that my changes are alright. Thank you for your time! GabeTucker (talk) 06:28, 14 January 2023 (UTC)

• David Eppstein? (Hate to put this on you, but I'm just too busy for a few days.) EEng 11:01, 15 January 2023 (UTC)

This complicates the explanation for no clear benefit. The older, simpler calculation is preferable. - MrOllie (talk) 02:52, 16 January 2023 (UTC)

"This complicates the explanation for no clear benefit" here is a list of the clear benefits my proof provides:

• My calculation is 23 lines, compared with the 21 lines of the original equation. So although it is negligibly longer, this allows us to more clearly convey the steps to arrive at our conclusions.
• The original calculation shows:

${\begin{aligned}::P(A,B,C)&=P(A|B,C)P(B,C)\\::&=P(B|A,C)P(A,C)\\::&=P(C|A,B)P(A,B)\\::&=P(A,B|C)P(C)\\::&=P(A,C|B)P(B)\\::&=P(B,C|A)P(A)::\end{aligned}}$

The majority of these lines play no role in the derivation and the explanation of Bayes' rule can be greatly simplified, like in my proof.

• The original calculation only calculates P(H3|C2,X1) = 2/3. This does not prove that P(H3|C1,X1) = 1/3, nor that P(H3|C3,X1) = 0.
• The original calculation shows the following premises:

${\begin{aligned}::P(H3|C1,X1)&={\tfrac {1}{2}}\\::P(H3|C2,X1)&=1\\::P(H3|C3,X1)&=0\\::P(Ci)&={\tfrac {1}{3}}\\::P(Ci,Xi)&=P(Ci)P(Xi)\\::P(H3|X1)&={\tfrac {1}{2}}::\end{aligned}}$

However, it does not justify any of these premises intuitively except for P(H3|X1) = 1/2, which it takes an entire two lines to do. My proof concisely justifies all of these premises in just three lines.

• Even so, most of these premises hold true for H3 and H3 alone. One intuitively understands the probability of a door being chosen given a car is behind it is 0. However, the premises only apply this rule to H3, which makes the explanation follow less directly from the rules defined using English (i.e., the rules state the host will open a door without a car behind it, not that door 3 is opened given door 1 has a car). Moreover, it does not provide intuition for turning the Monty Hall problem rules into general, stronger rules that cover all possibilities. Understanding how to model the core of where something comes from is healthy practice when trying to gain intuitions surrounding a complex topic.
• The original proof does not define the conditional probability identity, which is used in the derivation.

"The older, simpler calculation is preferable" This is incredibly vague. Can you please give any amount of specifics?

GabeTucker (talk) 03:25, 16 January 2023 (UTC)

@GabeTucker:

One should not write things like this:

P(C1|X1,H3)

Instead, that should look like this:

P(C₁ | X₁, H₃)

Notice that

• The letters are italicized but the digits and the parentheses and the vertical slash are not. This is codified in WP:MOSMATH. The point is to follow LaTeX style as closely as possible. One should also not italicize things like max, cos, log, sup, det, etc.
• Actual subscripts are used. (Doesn't that seem easier on the eyes to you?)
• Horizontal space precedes and follows the vertical slash.

Likewise the following

$P(H3|C1,X1)$

should instead look like this:

$P(H_{3}\mid C_{1},X_{1})$

Here again, actual subscripts are used. The vertical slash is coded as \mid.

Michael Hardy (talk) 20:06, 3 February 2023 (UTC)

Thank you for letting me know—I will remember this for the future, and I will integrate this into my solution should people decide my solution is worth implementing. GabeTucker (talk) 22:37, 20 March 2023 (UTC)

References

RfC about the proof

Are the changes proposed in the above proof reasonable for the direct calculation section of the Monty Hall problem page?

GabeTucker (talk) 11:39, 19 January 2023 (UTC)

EDIT: I have just updated the proof to no longer rely on mathematical prose, to include a citation, to improve the organizational structure, and to elaborate on issues that might be unclear to the reader (e.g., why P(Hi|Xj) = 1/2). These improvements address many of the issues raised in the comments.

Hi GabeTucker. I removed the RfC tag. Can we work together to craft a "brief and neutral" opening question? Better yet, could we post a neutrally-worded request for input at WP:WikiProject Statistics and WP:WikiProject Math, where we're more likely to get input from editors familiar with this topic area? Firefangledfeathers (talk / contribs) 17:32, 19 January 2023 (UTC)

@Firefangledfeathers Sure, I've neutralized and shortened my question. Are we good to readd the RfC tag? And how might I go about asking on the pages you provided? Thanks for the help. GabeTucker (talk) 17:46, 19 January 2023 (UTC)

I think that's much better. You don't need to link the article, as the central RfC listings will direct interested editors to this page. For the WikiProjects, you'd just start a new talk page section at WT:WPSTAT and/or WT:WPM and say something like "There's a dispute at Talk:Monty Hall problem#Update to direct calculation of Monty Hall problem about how best to present a proof. We're hoping interested editors can help us decide between two options." To be clear, I'd prefer to do this instead of opening an RfC, which I view to be one of the most expensive dispute resolution options in terms of community time, but you could do both if your determined to open an RfC. Firefangledfeathers (talk / contribs) 17:52, 19 January 2023 (UTC)

Thanks so much for the help! I'll post in one of those talk pages in lieu of opening an RfC. Again, I appreciate the help. GabeTucker (talk) 18:26, 19 January 2023 (UTC)

Both the current version and the proposed changes go against the general ethos that we are writing an encyclopedia, not a textbook, and aren't really in accord with the style we should adopt for mathematics prose. XOR'easter (talk) 22:00, 19 January 2023 (UTC)

Thanks for letting me know, I hadn't realized this was an issue. If I could spend some time on restructuring my changes such that they read more intuitively and less like a textbook/whiteboard lecture, would my changes be acceptable as an improvement to the current version? GabeTucker (talk) 23:05, 19 January 2023 (UTC)

Potentially; it's hard to say without a specific text to read, of course. And to be compliant with policy, we need at least one reference to a reliable source that works through the problem in this way and explains the virtues of doing so; we can't just assert our own opinions about what is most clarifying. XOR'easter (talk) 16:47, 21 January 2023 (UTC)

Hi there—thanks for the reply. How's it looking now that I updated it? GabeTucker (talk) 21:02, 23 January 2023 (UTC)

Looking at your changes, they don't seem to have addressed the issue of prose style. The text is still full of "we" and "you" colloquialisms, and it still contains unencyclopedic remarks like "We are in luck". XOR'easter (talk) 03:05, 26 January 2023 (UTC)

• (Summoned by bot) First, I see that the RfC tag was removed. That should not have been done. Please reinstate it. Second, someone should please state the issues in a manner understandable to the mathematically illiterate. Coretheapple (talk) 16:31, 21 January 2023 (UTC)

  Hi Coretheapple. I removed the RfC tag since the RfC was started with a non-brief, non-neutral statement, and the OP and I discussed alternatives for dispute resolution, including re-starting the RfC with a better opener. We decided to go with some WikiProject posts, which have (as far as I can tell) garnered the involvement of some experienced math/probability content area editors. I feel ok about my actions, and won't reinstate the RfC tag, but I also won't edit war over it if someone really feels an RfC is needed. Firefangledfeathers (talk / contribs) 19:06, 21 January 2023 (UTC)

  Bots often summon me to very poorly drafted RfCs. If you decide to go ahead with one, if you can't come to a consensus, I'd suggest that it be made clearer if possible. Coretheapple (talk) 21:34, 21 January 2023 (UTC)

  Will do! Firefangledfeathers (talk / contribs) 03:55, 22 January 2023 (UTC)

  Hi there—thanks for the reply. How's it looking now that I updated it? GabeTucker (talk) 21:03, 23 January 2023 (UTC)

Updated what? Please revise the beginning of this RfC. Remember that some of us can barely balance our checkbooks. Coretheapple (talk) 22:38, 23 January 2023 (UTC)

I updated the proof to no longer rely on mathematical prose, to include a citation, to improve the organizational structure, and to elaborate on issues that might be unclear to the reader (e.g., why P(Hi|Xj) = 1/2). GabeTucker (talk) 16:20, 24 January 2023 (UTC)

• Neither On further reflection, we should not include either option, as they are both unsourced / WP:OR. The section should just be removed entirely. - MrOllie (talk) 21:29, 23 January 2023 (UTC)

  Hi User:MrOllie—please take a look at my comment to User:Firefangledfeathers right below and let me know whether this source will be acceptable. GabeTucker (talk) 23:42, 23 January 2023 (UTC)

• Neither, mostly per comments by XOR and MrOllie. This is a problem about which we have an abundance of reliable sources. If this line of explanation isn't common, or at least present, in that body of sources, we shouldn't be spending so much article real estate on it. I'm open to changing my mind if such sources are cited. Firefangledfeathers (talk / contribs) 21:57, 23 January 2023 (UTC)

  Hi Firefangledfeathers—I based my answer on the general structure of the proof from a paper that I cited in my update: Gillman, Leonard (1 January 1992). "The Car and the Goats". The American Mathematical Monthly. 99 (1): 3–7. You can access it here: http://scipp.ucsc.edu/~haber/ph112/goats.pdf. I made some changes in order to improve concision and clarity since his proof was 6 pages long and had some more advanced statistics than most might understand upon a brief reading. But overall, it follows the same structure (sections 3, 5, and 6 are the most important for our purposes). With that in mind, is this update acceptable? GabeTucker (talk) 23:41, 23 January 2023 (UTC)

  Taking someone else's argument and then modifying it in the name of "concision and clarity" isn't what Wikipedia is suited for. XOR'easter (talk) 03:07, 26 January 2023 (UTC)

  Fair enough.

  I'll rewrite the entire argument along the lines of one of the mainstream proofs once I get the chance (without using prose style). Thanks for the input. GabeTucker (talk) 05:13, 26 January 2023 (UTC)

  Honestly, I don't see why the section exists. What benefit does it bring? Why is it not redundant with all the text above it? XOR'easter (talk) 21:24, 29 January 2023 (UTC)

  The Monty Hall problem is an article about statistics and probability. So computing a statistical solution to a problem about statistics and probability makes a lot of sense, at least on the surface. GabeTucker (talk) 06:30, 30 January 2023 (UTC)

  Isn't that what all the text above it does, multiple times? What does this section offer, besides more notation? XOR'easter (talk) 12:30, 30 January 2023 (UTC)

  I think the notation offers mathematical precision and clarity that a text description can't capture in the same way. At least, understanding it in this way helped me to better understand the topic. GabeTucker (talk) 11:29, 31 January 2023 (UTC)

  But it doesn't say anything that hasn't already been said. Whatever precision it offers is no better than what one would get from reading the prose carefully. The list of formulae is just a transcription of the scratch-work one would do in a notebook during such a careful reading. XOR'easter (talk) 15:28, 31 January 2023 (UTC)

  Is that really so? After looking through the article, I struggle to find a place that would have a one-to-one mapping, or even any rough mapping, with the provided Bayesian proof. GabeTucker (talk) 08:02, 1 February 2023 (UTC)

• Comment (Summoned by bot) I would respectfully suggest that all mathematical formulas not be in the article unless sourced. Coretheapple (talk) 19:57, 25 January 2023 (UTC)

• Comment I might be a little late to the subject, but I think the issue raised by GabeTucker for this article is really true: it lacks a formal demonstration written using standard notation. There is a reason that this notation exists, and it is precisely for clarity. The fact that you can somehow work out the formulas by knowing probability theory and transcribing from the article text does not mean that such a contribution would not improve greatly the article. We do not explain Matrix multiplication by only using text, even if it could be done. Likewise, WP:NOR does not preclude from reformulation. So I did not have the time yet to analyse the contribution in depth, but I think it should not be dismissed on the basis of the arguments that have been brought forward so far. Cochonfou (talk) 21:53, 14 March 2023 (UTC)

  Hi Cochonfou,

  I appreciate your comment, and I agree with your points regarding clarity and how WP:NOR does not preclude reformulation.

  An additional issue I've been encountering since the onset of this thread is the following: there are no reliable sources proving the Monty Hall Problem. The only sources proving the Monty Hall Problem do so informally and inconcisely, and are mostly done by non-academics writing blogs. So me writing a Bayesian proof for the Monty Hall Problem based on original research is just as credible as citing one of these blogs since in both cases, there are no academic credentials solving this proof. Examples:

  Top 3 sources after searching:

  • "Greek Data Guy": https://towardsdatascience.com/solving-the-monty-hall-problem-with-bayes-theorem-893289953e16#:~:text=Bayes%20Theorem%20%2B%20Monty%20Hall&text=Let's%20assume%20we%20pick%20door,correct%20if%20Monty%20opened%20B%20.

  This solution does not provide a Bayesian proof.

  • Julia Daikawa: https://medium.com/@judaikawa/the-monty-hall-problem-and-the-bayes-theorem-4415e50e233f

  This does provide a Bayesian proof for the Monty Hall Problem, but it does so A) using vague notation and B) without clearly establishing our premises. The source is non-academic, and it does not provide a proof for all three conditions.

  • Brilliant: https://brilliant.org/wiki/monty-hall-problem/

  This does not use standard notation, nor does it prove all three conditions. Moreover, it does not have a listed author, and its only source is non-academic and does not include the proof they provided.

  Academic sources:

  • MIT: http://web.mit.edu/rsi/www/2014/files/MiniSamples/MontyHall.Old.Bad/montymain.pdf

  This is a computer science student simulating the Monty Hall Problem without proving it.

  • Whitman College: https://www.whitman.edu/documents/Academics/Mathematics/2019/Lopez-Martinez-Keef.pdf

  This is an unpublished paper by an undergraduate student (non-credible source) proving the Monty Hall Problem.

  • Mathematics Magazine: https://www.researchgate.net/publication/233565559_The_Monty_Hall_Problem_Reconsidered

  This article proves alternative formulations of the Monty Hall Problem than the one in this Wikipedia article.

  I think the reason is pretty simple: no serious academics are going to be spending time to prove a relatively simple statistics problem, and no research article is going to publish this since it is, at this point, self-evident. There is nothing interesting to be said about a simple Bayesian explanation of the Monty Hall Problem by serious statisticians. So it becomes impossible for us to base our formulation of the Monty Hall Problem off a credible source. As a result of this, should we have no standard notation? On Cochonfou's point: clearly not, since standard notation provides clarity that text does not offer.

  Moreover, WP:NOR says there cannot be non-original "facts, allegations, and ideas". This seems to be a statement primarily about things in the world, like discoveries. In addition, this proof is not a point of view, so it does not break WP:NOR's neutrality rule. And, most importantly, it is verifiable, as it is a simple mathematical proof. Simple mathematical proofs do not seem like things that need credible sources when you are referencing other (cited) Wikipedia articles that establish the rules you are using in said proof.

  GabeTucker (talk) 23:37, 20 March 2023 (UTC)

Any real mathematicians ?

This article is lovely insofar as it illustrates so well confusion. It should be listed under "Magical Thinking", or "Cognitive bias", and probably "Vain attempts". I will not even begin to correct it, nor explain (I know my WP); just let possible startled readers know that yes, it is a lot of gobbledigook, some of it by people who are perfectly aware and enjoy it.Environnement2100 (talk) 10:55, 8 April 2023 (UTC)