Definition
Monty Hall problem is a mathematical brain teaser dealing with probabilistic decision making. It originated from a TV show hosted by Monty Hall in 1963. It is a very good example of how probabilistic scenarios may seem simple but yet at times can be difficult to wrap our minds around them. [1]
Who
This problem is recommended for anyone interested in statistics, especially Bayes Theorem. Enjoy!
What
The problem is simple, you go to Monty’s game show “Let’s make a deal”. There are 3 doors in front of you. Behind 1 of those doors, there is a car and behind the remaining two doors are goats! If you select the door behind which the car is there, you drive it home. The host asks you to select a door, you pick one. Then comes the fun part. After you picked a door, the host will open one of the two remaining doors to always reveal a goat. Now he would ask you, are you going to switch your selection or are you staying with your original choice? This is where the dilemma kicks in. [1]
Why
The solution to Monty Hall problem seems weird because our mental assumptions for solving the problem do not match the actual process. Our mental assumptions were based on independent, random events. However, Monty knows the prize location and uses this knowledge to affect the outcomes in a non-random fashion. Once you understand how Monty uses his knowledge to pick a door, the results make sense. [2]
The Monty Hall problem underscores a valuable lesson in probability theory: updating probabilities based on new information is a crucial aspect of making informed decisions. Whether in game shows or real-life situations, understanding how probabilities evolve as circumstances change can lead to more favorable outcomes. So, while the Monty Hall problem might have seemed counterintuitive at first, it serves as a compelling reminder that sometimes, probabilities can be much more intricate than they initially appear. [3]
| Car is in Door | Contestant Chooses Door | Monty Hall Opens Door | Contestant Switches | Result |
|---|---|---|---|---|
| A | A | B or C | A for B or C | Loses |
| A | B | C | B for A | Wins |
| A | C | B | C for A | Wins |
| B | A | C | A for B | Wins |
| B | B | A or C | B for A or C | Loses |
| B | C | A | C for B | Wins |
| C | A | B | A for C | Wins |
| C | B | A | B for C | Wins |
| C | C | A or B | C for A or B | Loses |
| Loses: 33% (1/3) Wins: 66% (2/3) |
See Theoretical Knowledge Vs Practical Application.
How
Many of the References and Additional Reading websites, and Videos will assist you with understanding the Monty Hall Problem.
As some professors say: “It is intuitively obvious to even the most casual observer.”
References
[1] “Monty Hall Problem — An Empirical Proof”. 2021. Medium. https://towardsdatascience.com/monty-hall-problem-an-empirical-proof-60e7e0761503.
[2] Frost, Jim. 2017. “The Monty Hall Problem: A Statistical Illusion”. Statistics By Jim. https://statisticsbyjim.com/fun/monty-hall-problem/.
[3] ⭐ “Demystifying the Monty Hall Problem: A Step-by-Step Guide.” 2023. Dylan Tintenfich. August 22. https://dylannalex.github.io/monty_hall/.
Additional Reading
⭐ Ali. 2024. “What Happened to the World’s Smartest Woman After She Solved a Math Question?” Medium. Medium. July 26. https://ali.medium.com/what-happened-to-the-worlds-smartest-woman-after-she-solved-a-math-question-019943967383.
Apart from these intricate puzzles, many mathematical problems have captured the public’s attention throughout the years. One of the most famous among them is the Monty Hall problem, a delightful probability question that mesmerises both casual fans and seasoned mathematicians. What sets the Monty Hall problem apart from others is its clear-cut solution, which many find hard to believe. This is primarily because the answer defies our natural instincts; it feels so contrary to what we would intuitively think that even today, many struggle to accept the correct resolution. The Monty Hall problem not only challenges our understanding of probability but also invites us to reconsider how we think about choices and chance.
Dutta, Samrat. “The Monty Hall Mystery: A Tale of Prizes and Probability”. 2023. Medium. https://medium.com/predict/the-monty-hall-mystery-a-tale-of-prizes-and-probability-bae5eef6b510.
The Monty Hall Problem serves as an enchanting window into the world of probability, questioning our perceptions and revealing the power of fine logic. It highlights the significance of critical thinking and understanding probabilities.
Huber, Mark. “The Monty Hall Problem”. 2013. www1.cmc.edu. https://www1.cmc.edu/pages/faculty/MHuber/Research/talks/huber_talk_2013d.pdf.
“Marilyn Vos Savant | The Game Show Problem (Printer-Friendly)”. 2022. web.archive.org. https://web.archive.org/web/20100928000559/http://marilynvossavant.com/articles/gameshow_print.html?t=64.
“Monty Hall Problem: Solution Explained Simply.” 2024. Statistics How To. May 12. https://www.statisticshowto.com/probability-and-statistics/monty-hall-problem/.
“The Monty Hall Problem”. 2021. Medium. https://www.cantorsparadise.com/the-monty-hall-problem-6e3e56e454a2.
“The Monty Hall Problem”. 2019. Medium. https://medium.com/swlh/the-monty-hall-problem-3383e3048aa8.
“Monty Hall Problem | Brilliant Math & Science Wiki”. 2022. brilliant.org. https://brilliant.org/wiki/monty-hall-problem/.
“Monty Hall Problem | Understand Monty Hall Problem In Detail”. 2019. EDUCBA. https://www.educba.com/monty-hall-problem/.
“The Monty Hall Problem – Math Images”. 2022. mathimages.swarthmore.edu. https://mathimages.swarthmore.edu/index.php/The_Monty_Hall_Problem.
“Monty Hall Problem – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Monty_Hall_problem.
“🐐 Monty Hall Problem Simulator”. 2022. montyhall.io. https://montyhall.io/.
⭐ “Understanding the Monty Hall Problem.” 2024. BetterExplained. Accessed June 3. https://betterexplained.com/articles/understanding-the-monty-hall-problem/.
Here’s the key points to understanding the Monty Hall puzzle:
1. Two choices are 50-50 when you know nothing about them
2. Monty helps us by “filtering” the bad choices on the other side. It’s a choice of a random guess and the “Champ door” that’s the best on the other side.
In general, more information means you re-evaluate your choices.
The fatal flaw in the Monty Hall paradox is not taking Monty’s filtering into account, thinking the chances are the same before and after. But the goal isn’t to understand this puzzle — it’s to realize how subsequent actions & information challenge previous decisions. Happy math.
Vazsonyi, Andrew. “Which Door Has the Cadillac?” 2022. web.archive.org. https://web.archive.org/web/20140413131827/http://www.decisionsciences.org/DecisionLine/Vol30/30_1/vazs30_1.pdf.
Videos
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⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
The featured image on this page is from the The Monty Hall Problem page on the Medium website.
Mathematical Mysteries 