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Far East Journal of Mathematical Education

MONTY HALL PROBLEM IN UNINTENTIONAL RANDOM MEASUREMENTS

Shiro Ishikawa

Published Sep 9, 2009• 0 citations
Publisher

Abstract

Recently we proposed measurement theory, which is a kind of quantitative language for the descriptions of ordinary science, and is based on the quantum mechanical world view. The purpose of this paper is to clarify the Monty Hall problem in measurement theory. For this, we first introduce partial statistical measurements and unintentional random measurements. And further, we clarify the principle of equal probability (i.e., the probabilistic interpretation of no information). Under these preparations, we completely understand the Monty Hall problem.

DOI
10.17654/0973563109014

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Recently we proposed measurement theory, which is a kind of quantitative language for the descriptions of ordinary science, and is based on the quantum mechanical world view. The purpose of this paper is to clarify the Monty Hall problem in measurement theory. For this, we first introduce partial statistical measurements and unintentional random measurements. And further, we clarify the principle of equal probability (i.e., the probabilistic interpretation of no information). Under these preparations, we completely understand the Monty Hall problem.
In this paper, we study the principle of equal probability (i.e., unless we have sufficient reason to regard one possible case as more probable than another, we treat them as equally probable) in measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics with the Copenhagen interpretation. This turn from physics to language does not only realize theremarkable extensionof quantum mechanicsbut alsoestablish the method of science. Our study will be executed in the easy example of the Monty Hall problem. Although our argument is simple, we believe that it is worth pointing out the fact that the principle of equal probability can be, for the first time, clarified in measurement theory (based on the dualism) and not the conventional statistics (based on Kolmogorov’s probability theory).
The Monty Hall problem is a classic probability puzzle known for its counterintuitive solution, revealing fundamental discrepancies between mathematical reasoning and human intuition. To bridge this gap, we introduce a novel explanatory framework inspired by quantum measurement theory. Specifically, we conceptualize the hosts' actions-opening doors to reveal non-prizes-as analogous to quantum measurements that cause asymmetric collapses of the probability distribution. This quantum-inspired interpretation not only clarifies why the intuitive misunderstanding arises but also provides generalized formulas consistent with standard Bayesian results. We further validate our analytical approach using Monte Carlo simulations across various problem settings, demonstrating precise agreement between theoretical predictions and empirical outcomes. Our quantum analogy thus offers a powerful pedagogical tool, enhancing intuitive understanding of conditional probability phenomena through the lens of probability redistribution and quantum-like measurement operations.
Endeavoring to formulate an exhaustive solution to the measurement problem in view of the theory of decoherence leads to a better understanding of the status of the collapse and of the emergence of classicality, thanks to a precise definition of the measurement and some new vocabulary to speak about quantum mechanics. Considering the latter as a probabilistic theory all along allows us to avoid the usual probability problem of the many-worlds interpretations. A thorough verification of the consistency of quantum mechanics at all scales is proposed, as well as a discussion of what can be deemed an observer.
The aim of this paper is to consider the consequences of an information-theoretic interpretation of quantum mechanics for the measurement problem. The motivating idea of the interpretation is that the relation between quantum mechanics and the structure of information is analogous to the relation between special relativity and the structure of space-time. Insofar as quantum mechanics deals with a class of probabilistic correlations that includes correlations structurally different from classical correlations, the theory is about the structure of information: the possibilities for representing, manipulating, and communicating information in a genuinely indeterministic quantum world in which measurement outcomes are intrinsically random are different than we thought. Part of the measurement problem is deflated as a pseudo-problem on this view, and the theory has the resources to deal with the remaining part, given certain idealizations in the treatment of macrosystems.
Recently we proposed the linguistic interpretation of quantum mechanics (called quantum and classical measurement theory, or quantum language), which was characterized as a kind of metaphysical and linguistic turn of the Copenhagen interpretation. This turn from physics to language does not only extend quantum theory to classical systems but also yield the quantum mechanical world view (i.e., the philosophy of quantum mechanics, in other words, quantum philosophy).And we believe that this quantum language is the most powerful language to describe science. The purpose of this paper is to describe the Monty-Hall problem and the three prisoners problem in quantum language. We of course believe that our proposal is the final solutions of the two problems. Thus in this paper, we can answer the question: "Why have philosophers continued to stick to these problems?" And the readers will find that these problems are never elementary, and they can not be solved without the deep understanding of "probability" and "dualism". KEY WORDS: Philosophy of probability, Fisher Maximum Likelihood Method, Bayes' Method,The Principle of Equal (a priori) Probabilities