For my research, I will be taking a deeper look into “The Monty Hall Problem.” This idea arises from the old game show titled “Let’s Make a Deal.” At the end of the show, a contestant would have an opportunity to pick from one of three doors. One door has a car behind it. The other two have goats. The contestant would pick one of the three doors. Then, the host, Monty Hall, would reveal one of the goats behind a door that was not picked. Now, the contestant needs to choose to either keep the door chosen, or move to the other unopened door.
Now, it may seem as if this is a pointless part of the game to add suspense. In reality, maybe, but it can work out in favor of the contestant just about every time. When the contestant picked a door the first time, their odds of choosing the car were 1/3. By removing a door, it would seem as if the chance has dropped down to a 50/50 shot of winning the car. However, it is proven that that is not the case. The sources below will be used to show exactly how switching doors increases the chance of winning a brand new car:
This article explains the “Three Prisoner’s Problem.” although a different premise is introduced, this problem is the same exact thing as the Monty hall problem. Three prisoners are jailed and two are to be executed at random. The warden is the only person who knows who walks. Prisoner A asked the warden if he would tell him the name of at least one other prisoner who was to be executed. The warden responds with Prisoner B. Prisoner A tells Prisoner C what he found out and Prisoner C realized that he now has a 2/3 chance at survival, while Prisoner A is left with a 1/3 chance. The explanation is roughly the same.
How I Intend to Use it: If the first explanation of the problem leaves the reader confused, I believe relating it to this second problem may help. It gives both problems a base for each other to stand on. This could make it easier for the reader to grasp the premise and understand the solution to the problem introduced as my counterintuitive topic.
This article, (Despite coming from Wikipedia,) explains a mathematical theory knows as Bayes’ Theorem. Bayes’ Theorem is essentially the mathematical equation explaining how the Monty Hall problem works. The theorem looks like this:
P[A|B] = {P[B|A]P[A]}/P[B] = {P[B|A]P[A]}/P[B|A]P[A]+P[B|C]P[C]
Basically, the odds start at 1:1:1. Lets say the contestant picks door A and Monty opens door C. The ratio changes to 1:2:0. Now, given the option to switch to door B, it would be wise for the contestant to take it since the ratio shows it has the highest chance of containing the car.
Bayes’ Theorem is the explanation behind the problem.
How I Plan to Use it: Up to this point, I have explained why the problem works the way it does. With Bayes’ Theorem, I can show why this is how it works. Bayes’ Theorem is the foundation to the understanding of the problem.
This article touches on the Monty Hall problem. It is not nearly as in depth as the others, but it does offer another way to look at the solution. Supposed the contestant picks door A, which is the door that the car is behind. If doors are switched, loss is certain. However, if the contestant picked door B, the 1 goat would be revealed and if she switched, she would win the car. Same goes for door C. when looking at it this way, you have a 2/3 chance to win as well.
How I Plan to Use it: The article gives a second reason to why Monty Hall problem works the way it does. It lays out each and every situation the contestant could be faced with. In the end, out of the three situations, the contestant would only lose one time and would win the other two providing she switched. I will use this information to introduce another way of looking at the solution to the problem by mapping it out rather than explaining the probability.
Marilyn Vos Savant – Game Show Problem
This article shows a timeline of Marilyn Vos Savant’s battle with the public on the logic of this problem. This article shows how so many different students as well as mathematicians were against her logic when she first explained the problem and the response she gave to make sense of it all. She also gives examples and experiments which test the hypothesis that switching wins more. This article clarifies the base argument that swapping to the side with more doors is beneficial.
How I Plan to Use it: This article provides all the base to my topic. It explains the background of how the problem came to be and where the idea came from. It also states the exact original problem as asked by somebody. Savant provides the solution of why the problem works the way it does in great detail. she even provides an experiment for readers to try and invites them to report back to her.
Understanding the Monty Hall Problem
This article provides not only a simulation to work with, but also another example to better understand why switching doors is better. this example contains the same problem, but uses 100 doors rather than 3.
How I Plan to Use it: In this article, the author explains the problem with a slight twist. Say that instead of three doors, there are 100. the contestant picks one, and Monty opens up 98/99 of the doors the contestant had not picked. Think the contestant would start feeling the pressure if he did not switch? It is the same exact problem as the un-altered version, but it emphasizes the purpose of switching doors, and I would like to use that as a good example of why to switch
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My Research:
statistics out of 200:
traditional, Always switch doors:
Wins: 125
Losses: 75
Win %: 62.50
Statistics out of 100:
out of 100 doors, Always switch doors:
wins: 100
Losses: 0
Win %: 100.00%
traditional, never switch:
Wins: 34
Losses: 66
Win %: 34.00
Traditional, switch half of the time:
Wins: 50
Losses: 50
Win %: 50.00
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Counterintuitive Fall 2022 
Professor, if you have the time to leave me any feedback, i would really appreciate it.
~Vinny
Feedback provided. —DSH
This is nice work, Vinny. Your link to “Understanding . . . .” is broken but I found the article by searching the title at betterexplained.com
I must say, the explanation at betterexplained did me very little good. Marilyn vosSavant’s chart, however, made the “odds” problem completely obvious.
I’d suggest you use the term “50/50 odds” as the topic of your definition essay, Vinny. Clearly the two items in a pair are sometimes identical, sometimes not. The question: is David or is Vinny more likely to win tonight’s lottery is a 50/50 proposition every time unless we know Vinny bought a ticket.
What’s most intriguing about the problem is that the difference-maker is what Monty (or the warden) knows! The fact that he will never reveal the car forces his hand 2 out of 3 times. For me, that’s the “hidden” solution or explanation. If Monty opens a door at random, not knowing what’s back there, and finds a goat, the contestant might as well stick with her original choice. Right?
The great value of the betterexplained post was its 3-door engine. I’m delighted you were able to do your own research so conveniently. On the other hand, watching the light slowly dawn on experimenters would be priceless. Do you know anybody well enough to convince them to do 100 trials with physical objects?
I have an idea you could use to supplement your argument, Vinny, or perhaps even change the emphasis of your paper entirely. The Monty Hall problem has been pretty much solved by now, and the best you might be able to accomplish with it is to demonstrate its inherent counterintuitivity and then resolve the confusion in your reader’s mind; in effect, providing a bit of education.
But what if, instead, you examined the possibilities not considered by the TV version of the game? After all, the odds are only improved in the Classic Version because Monty offers us a choice every time. Suppose, without the contestants’ knowledge, Monty followed the rule that he only offered a choice when the contestant had chosen the car? That would change everything.
It also reveals the problem inherent in trying to live our lives like a game show. We never truly know the rules well enough to predict whether what we’re being offered will pay off or not.
Please fix your pronouns, Vinny. They’re costing you half a grade now because I’ve corrected them so many times without effect.
1. Never use “you.”
2. Keep your numbers straight; don’t mix singular and plural. (Never say: would THE CONTESTANT feel the pressure if THEY didn’t switch).
Professor, i m just going to respond to all of your comments here.
The betterexplained article is rather useless i know. To be honest it gave me a small amount of new information i could include but i had no idea what else i could have possibly needed to help explain this problem. Maybe i will swap them out with something similar to the riddle you introduced in class yesterday, i’m not sure yet.
I like the idea of 50/50 odds as my topic for the definition essay. I had not thought that odds are not necessarily what they are, but what you know. Maybe i can dive deeper into it with my post this weekend.
I may turn the tone of my paper towards a research rather than a summary. I would just be worried i would strafe from my topic too much with the new material and lose sight of the actual counter intuitive piece my paper is supposed to be about. What do you think?
I believe i have fixed mistakes in here.
If you have the time, would you look over it again and provide more feedback if needed? Thank you!
Happy to help if I can, Vinny.
The first thing I notice is very little pronoun trouble in the Introduction. (Maybe there was very little there the first time; I don’t look back.) But one does remain. These are difficult to spot if you don’t naturally “feel” them, which is why I’m trying to train you to sense them for yourself. Your singular contestant chooses “their door” and improves “their odds.” Both are plural. Two solutions as always, are: 1) stick to plurals (contestants improve their odds), or 2) eliminate the pronouns altogether (when the contestant picked a door, the odds were 1/3).
“You” language remains in “Getting the Goat.”
You claim that the Bayesian formula will clear things up for your readers, but do you believe it? I find hard thinking more productive than that formula, which for me does not clarify anything.
This source http://www.ece.umd.edu/~tretter/enee324/threeprisoners.pdf offers some nice insight into the importance of knowing the query that produced the answer the guard gave.
By the way, maybe I’m reading it wrong, but the two versions of the prisoner problem articulated by Daniel Dendy seem to me to produce the same results but hopelessly cloud A’s understanding much more than necessary. When B is named, A has no way of knowing whether it’s because C is to be pardoned or whether A is to be pardoned.
This questions of how the choices are made and how the information about the situation is obtained seem to be at the bottom of the entire paradox much more than the bare numbers of the possibilities, don’t they?
It’s also funny to consider that, after A (who initially chooses to be A) learns that B will be executed (he’s a goat), a clear-thinking A would desperately wish to become C (with a now 2/3 chance of being a car).
Alright, I revised the essay to reflect pronoun changes and remove the “You” language in “Getting the Goat.”
While i don’t believe the Bayes’ theorum itself (the formula) helps all too much, i believe the explanation i had put right under it makes the understanding significantly easier. that being said, there are other ways in my sources that may give explanations with even more clarity than this.
I think the fact that A has no idea who is to be pardoned is important for the explanation. It shows how similar both stories are, the logic, but it leaves out a legitimate answer. The answer to the problem could be either C or A and i feel that if an answer was given, would unintentionally give the reader a biased answer when answering this problem, rather than leave it open for him to interpret himself.
I agree, the paradox is all just a method to solve a specific problem, given all the information received.
I didn’t think of that, that’s actually really funny.
——
Also, i figured i should explain this so i can stop doing it. My problem with using their is that i see it as a describing a possessive singular without hinting any gender. Not sure if that makes any sense you but what would i use instead of that?
Understandably i can always use he/his or she/hers but I’m wondering if there is a word that does the same thing without describing gender
Ex: The contestant showed that (Their) door had the car behind it.
Eventually, the language will evolve enough so that we no longer worry about the misuse of “their,” Vinny. Already it’s used so commonly that only English teachers even hear the error. You can use it at will in conversation and informal writing (and the history of English offers plenty of evidence that good writers have gotten away with it for centuries), but it’s still forbidden in formal writing. And no, there’s no good substitute for it.
The only legal formal alternatives to your example are:
The contestant showed that her door had the car behind it.
The contestant showed that his door had the car behind it.
The contestant showed that the chosen door had the car behind it.
If you’re planning to write in the singular (the contestant), choose a gender, or avoid a gender with passives, or ask why you’re bothering with a pronoun in the first place. MANY can be eliminated.
WITH: The contestant was disappointed when the door SHE selected didn’t win HER a car but only got HER a goat.
WITHOUT: The contestant was disappointed to win a goat instead of a car.
Thanks professor. I won’t make that mistake again