scholarly journals One game show, two boys, two aces, three prisoners - what’s an AI to do?

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Eric Neufeld ◽  
Sonje Finnestad
Keyword(s):  
Probability Theory ◽  
Word Problem ◽  
Random Variables ◽  
Game Show ◽  
Causal Diagram ◽  
Monty Hall ◽  
Monty Hall Problem ◽  
Technical Terms

We review a quartet of widely discussed probability puzzles – Monty Hall, the three prisoners, the two boys, and the two aces. Pearl explains why the Monty Hall problem is counterintuitive using a causal diagram. Glenn Shafer uses the puzzle of the two aces to justify reintroducing to probability theory protocols that specify how the information we condition on is obtained. Pearl, in one treatment of the three prisoners, adds to his representation random variables that distinguish actual events and observations. The puzzle of the two boys took a perplexing twist in 2010. We show the puzzles have similar features, and each can be made to give different answers to simple queries corresponding to different presentations of the word problem. We offer a unified treatment that explains this phenomenon in strictly technical terms, as opposed to cognitive or epistemic.  

scholarly journals Sleeping Beauty on Monty Hall

Philosophies ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 15
Author(s):  
Michel Janssen ◽  
Sergio Pernice
Keyword(s):  
Simple Model ◽  
Philosophy Of Science ◽  
Sleeping Beauty ◽  
Simple Solution ◽  
Degrees Of Belief ◽  
Sleeping Beauty Problem ◽  
Game Show ◽  
Bayes’S Theorem ◽  
Monty Hall ◽  
Monty Hall Problem

Inspired by the Monty Hall Problem and a popular simple solution to it, we present a number of game-show puzzles that are analogous to the notorious Sleeping Beauty Problem (and variations on it), but much easier to solve. We replace the awakenings of Sleeping Beauty by contestants on a game show, like Monty Hall’s, and increase the number of awakenings/contestants in the same way that the number of doors in the Monty Hall Problem is increased to make it easier to see what the solution to the problem is. We show that these game-show proxies for the Sleeping Beauty Problem and variations on it can be solved through simple applications of Bayes’s theorem. This means that we will phrase our analysis in terms of credences or degrees of belief. We will also rephrase our analysis, however, in terms of relative frequencies. Overall, our paper is intended to showcase, in a simple yet non-trivial example, the efficacy of a tried-and-true strategy for addressing problems in philosophy of science, i.e., develop a simple model for the problem and vary its parameters. Given that the Sleeping Beauty Problem, much more so than the Monty Hall Problem, challenges the intuitions about probabilities of many when they first encounter it, the application of this strategy to this conundrum, we believe, is pedagogically useful.

The Monty Hall problem

2019 ◽  
pp. 121-124
Author(s):  
Steve Selvin
Keyword(s):  
Game Show ◽  
Monty Hall ◽  
Monty Hall Problem

A famous problem that arose from a television game show that produced issues that are widely debated, called the “Monty Hall problem.”

Optimal strategies for the progressive Monty Hall problem

2009 ◽  
Vol 93 (528) ◽  
pp. 410-419
Author(s):  
Stephen K. Lucas ◽  
Jason Rosenhouse
Keyword(s):  
Optimal Strategies ◽  
The Other ◽  
Game Show ◽  
Monty Hall ◽  
Monty Hall Problem

In the classical Monty Hall problem you are a contestant on a game show confronted with three identical doors. One of them conceals a car while the other two conceal goats. You choose a door, but do not open it. The host, Monty Hall, now opens one of the other two doors, careful always to choose one he knows to conceal a goat. You are then given the options either of sticking with your original door, or switching to the other unopened door. What should you do to maximise your chances of winning the car?

Connecting Research to Teaching: Good Things Always Come in Threes: Three Cards, Three Prisoners, Three Doors

2006 ◽  
Vol 99 (6) ◽  
pp. 401-405
Author(s):  
Laurie H. Rubel
Keyword(s):  
Game Show ◽  
Monty Hall ◽  
Monty Hall Problem

Readers are likely to be familiar with the infamous Monty Hall problem, played on the Let's Make a Deal game show and later addressed in the “Ask Marilyn” column in a 1990 issue of Parade.

scholarly journals Monty Hall three door ’anomaly’ revisited: a note on deferment in an extensive form game

2021 ◽  
Author(s):  
Philipp E. Otto
Keyword(s):  
Probability Theory ◽  
Decision Problems ◽  
Extensive Form ◽  
Form Game ◽  
Information Updating ◽  
Extensive Form Game ◽  
Monty Hall ◽  
Initial Choice

AbstractThe Monty Hall game is one of the most discussed decision problems, but where a convincing behavioral explanation of the systematic deviations from probability theory is still lacking. Most people not changing their initial choice, when this is beneficial under information updating, demands further explanation. Not only trust and the incentive of interestingly prolonging the game for the audience can explain this kind of behavior, but the strategic setting can be modeled more sophisticatedly. When aiming to increase the odds of winning, while Monty’s incentives are unknown, then not to switch doors can be considered as the most secure strategy and avoids a sure loss when Monty’s guiding aim is not to give away the prize. Understanding and modeling the Monty Hall game can be regarded as an ideal teaching example for fundamental statistic understandings.

scholarly journals On Distribution Characteristics of a Fuzzy Random Variable

2018 ◽  
Vol 47 (2) ◽  
pp. 53-67 ◽  
Author(s):  
Jalal Chachi
Keyword(s):  
Probability Theory ◽  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Distribution Characteristics ◽  
Fuzzy Random Variables ◽  
Fuzzy Random

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

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