Applied Mathematics, 2012, 3, 788-794 http://dx.doi.org/10.4236/am.2012.37117 Published Online July 2012 (http://www.SciRP.org/journal/am) Monty Hall Problem and the Principle of Equal Probability in Measurement Theory Shiro Ishikawa Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan Email: [email protected] Received April 28, 2012; revised May 28, 2012; accepted June 5, 2012 ABSTRACT 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 quan- tum 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 the remarkable extension of quantum mechanics but also establish 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). Keywords: Linguistic Interpretation; Quantum and Classical Measurement Theory; Philosophy of Statistics; Fisher Maximum Likelihood Method; Bayes’ Theorem 1. Introduction (cf. Figure 1: ③ later), constructed as the scientific the- ory formulated in a certain C*-algebra A (i.e., a norm 1.1. Monty Hall Problem closed subalgebra in the operator algebra BH com- The Monty Hall problem is well-known and elementary. posed of all bounded operators on a Hilbert space H, cf. Also it is famous as the problem in which even great [14,15]). MT is composed of two theories (i.e., pure mathematician P. Erdös made a mistake (cf. [1]). The measurement theory (or, in short, PMT] and statistical Monty Hall problem is as follows: measurement theory (or, in short, SMT). That is, it has Problem 1 [Monty Hall problem 1]. You are on a the following structure: game show and you are given the choice of three doors. (A) MT (measurement theory) Behind one door is a car, and behind the other two are A:PMT1 goats. You choose, say, door 1, and the host, who knows where the car is, opens another door, behind which is a pure measurement causality goat. For example, the host says that AxiomP 1 Axiom2 (♭) the door 3 has a goat. And further, He now gives you the choice of sticking with door 1 or switching to door 2? What should you do? A:SMT In the framework of measurement theory [2-12], we 2 statistical measurement + causality shall present two answers of this problem in Sections 3.1 and 4.2. Although this problem seems elementary, we S Axiom 1 Axiom2 assert that the complete understanding of the Monty Hall problem can not be acquired within Kolmogorov’s prob- where Axiom 2 is common in PMT and SMT. For com- ability theory [13] but measurement theory (based on the pleteness, note that measurement theory (A) (i.e., (A ) dualism). 1 and (A2)) is not physics but a kind of language based on “the (quantum) mechanical world view”. As seen in [9], 1.2. Overview: Measurement Theory note that MT gives a foundation to statistics. That is, As emphasized in refs. [7,8], measurement theory (or in roughly speaking, short, MT) is, by a linguistic turn of quantum mechanics (B) it may be understandabl e to consider that PMT and Copyright © 2012 SciRes. AM S. ISHIKAWA 789 Figure 1. The development of the world views from our standing point. For the explanation of (①-⑧), see [8,10]. SMT is related to Fisher’s statistics and Bayesian statis- f 1 . The bi-linear functional f is also de- tics respectively. CΩ Also, for the position of MT in science, see Figure 1, noted by , f , or in short , f . CΩ * CΩ which was precisely explained in [8,10]. When A BH, the C*-algebra composed of all * c Define the mixed state C Ω such that compact operators on a Hilbert space H, the (A) is called * 1 and f 0 for all fC Ω such that quantum measurement theory (or, quantum system the- CΩ ory), which can be regarded as the linguistic aspect of f 0 . And put quantum mechanics. Also, when A is commutative (that * is, when A is characterized by C Ω , the C -algebra m ** 0 SCΩΩ C is a mixed state . composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff Also, for each Ω , define the pure state space Ω (cf. [16])), the (A) is called classical meas- SCm Ω * such that urement theory. Thus, we have the following classifica- * , fffC Ω. And put tion: CΩ CΩ quantum MT when A BH c p ** (C) MT SC ΩΩ C is a pure state , classical MT when ACΩ 0 which is called a state space. Note, by the Riesz theorem The purpose of this paper is to clarify the Monty Hall * problem in the classical PMT and classical SMT. (cf. [16]), that CM(Ω)(Ω) is a signed meas- ure on Ω and SCmmΩΩ* M is a meas- 2. Classical Measurement Theory (Axioms 1 and Interpretation) ure on Ω such that Ω 1 . Also, it is clear that 2.1. Mathematical Preparations SCp Ω * is a point measure at Ω , 00 0 Since our concern is the Monty Hall problem, we devote where ff d fCΩ . This im- 0 0 ourselves to classical MT in (C). Throughout this paper, Ω we assume that Ω is a compact Hausdorff space. Thus, plies that the state space SCp Ω * can be also iden- we can put C0 Ω CΩ, which is defined by a Ba- * nach space (or precisely, a commutative C -algebra) tified with Ω (called a spectrum space or simply, spec- composed of all continuous complex-valued functions on trum) such as a compact Hausdorff space Ω , where its norm f CΩ p * ∋ SC ΩΩ is defined by max f . Let C Ω * be the dual Ba- (1) Ω state space spectrum * nach space of C Ω . That is, C Ω is a con- Also, note that C Ω is unital, i.e., it has the identity I (or precisely, I ), since we assume that Ω is com- tinuous linear functional on C Ω , and the norm CΩ pact. * is defined by sup ff: CΩ such that According to the noted idea (cf. [17]) in quantum me- CΩ Copyright © 2012 SciRes. AM 790 S. ISHIKAWA chanics, an observable OXBF:,, X in C Ω is ciated. 2 defined as follows: (E2) For every tt, T, a Markov operator X 12 (D1) [Field] X is a set, BX (2 , the power set of X) Φ : CΩ CΩ is defined (i.e., Φ 0, is a field of X, that is, “ 12, BBXX1 2 ”, tt12, t2 t1 tt12, “ ΞΞBXXX B”. Φ I I). And it satisfies that ΦΦ (D2) [Additivity] F is a mapping from B to C Ω tt12, CΩCΩ tt23,,tt13 X tt21 satisfying: 1): for every Ξ B , F Ξ is a non-nega- X holds for any tt, , tt, T2 . tive element in C Ω such that 0 F Ξ I , 2): 12 23 F 0 and F X I, where 0 and I is the 0-ele- The family of dual operators ment and the identity in C Ω respectively. 3): for any ** * mm Ξ , Ξ B such that ΞΞ , it holds that Φtt, : SCΩΩ t SC t 1 2 X 12 12 1 2 2 tt12, T FFΞΞ12 Ξ 1FΞ 2. For the more precise argument (such as countably ad- is called a dual causal relation (due to the Schrödinger ditivity, etc.), see [7,9]. picture). When ** * pp 2.2. Classical PMT in (A1) ΦΩtt, SC t SC Ω t 12 1 2 In this section we shall explain classical PMT in (A1). 2 * holds for any tt, T, the causal relation is said to With any system S, a commutative C -algebra C Ω 12 be deterministic. can be associated in which the measurement theory (A) Here, Axiom 2 in the measurement theory (A) is pre- of that system can be formulated. A state of the system S sented as follows: is represented by an element SCp Ω * and an Axiom 2 [Causality]. The causality is represented by a causal relation Φ : CCΩΩ . observable is represented by an observable tt12, t 2t 1 2 tt12, T OXBF:,, in C Ω . Also, the measurement of * X For the further argument (i.e., the W -algebraic formu- the observable O for the system S with the state is lation) of measurement theory, see Appendix in [7]. denoted by MOS, or more precisely, CΩ 2.3. Classical SMT in (A2) MO:,,, XBFS . An observer can obtain CΩ X It is usual to consider that we do not know the state 0 a measured value x X by the measurement when we take a measurement MOS, . That is C MO, S. 0 CΩ P because we usually take a measurement MO , S The Axiom 1 presented below is a kind of mathe- C 0 matical generalization of Born’s probabilistic interpreta- in order to know the state . Thus, when we want to tion of quantum mechanics. And thus, it is a statement 0 without reality. emphasize that we do not know the the state , 0 AxiomP 1 [Measurement]. The probability that a mea- sured value x X obtained by the measurement MOS, is denoted by M OS, .