The Final Solutions of Monty Hall Problem and Three Prisoners Problem

1 The Final Solutions of Mon t y Hall P roblem and Three Prisoners Problem Shiro Ishik a w a Dep artment of Mathemat ics, F aculty of Scienc e and T e c hnolo gy, Ke io University, 3-14-1 Hiyoshi koho ku-ku, Y okohama, 223-852 2 Jap an. E-mail: ishikawa@math .keio.ac.jp Abstract Recen tly we prop osed the linguistic in terpr etation of quan tum mec hanics (called quan tum and classical measuremen t theory , or quantum language) , whic h w as c haracterized as a kind of meta- physic al and linguistic turn of the Cop enhagen in terpr etation. This turn from physic s to language do es not only extend quantum th eory to classical s y s tems bu t also yield the quant um mec hanical w orld view (i.e., the p hilosophy of quan tum mec hanics, in other w ords, quan tum ph ilosoph y).And w e b eliev e that this quan tum language is the most p o werful language to describ e science. Th e purp ose of this pap er is to describ e the Mo n t y-Hall p roblem and the three prisoners p r oblem in quan tum language. W e of course b eliev e that our pr op osal is the final solutions of the t w o problems. Th u s in this p ap er, w e ca n ans wer the questio n: ”Wh y ha ve philosopher s co n tin u ed to stic k to these p r oblems?”And the r eaders will find that these problems are n ev er elemen tary , and they can not b e solv ed without the deep u nderstand in g of ”probabilit y” and ”dualism”. Keyw ords : Philosoph y of probabilit y , Fisher Maxim u m Like liho o d Metho d, Ba yes’ Metho d, Th e Principle of Equal (a priori) Probab ilities 1 In tro duction 1.1 Mon ty H all problem and Three prisoners problem According to ref. [4], w e shall introd uce the usual descriptions of the M on ty H all problem and the three p risoners p roblem as f ollo ws. Problem 1 [Mon t y Hal l problem]. Supp ose y ou are on a game sho w, and y ou are giv en the c hoice of th ree d o ors (i.e., “Do or A 1 ” , “Door A 2 ” , “Door A 3 ” ). Behind one do or is a car, b ehind the others, goats. Y ou do not kn o w what’s b ehind the do ors Ho wev er, y ou pick a do or, sa y ”Door A 1 ”, and the host, who knows w hat’s b ehind the doors, op ens another do or, s ay “Do or A 3 ” , wh ich has a goat. He sa ys to y ou, “Do y ou wa n t to pic k Do or A 2 ?” Is it to y our adv an tage to switc h yo ur c hoice of do ors? ❄ ❄ ❄ Do or A 1 Do or A 2 Do or A 3 Problem 2 [T h ree prisoners problem]. Three prisoners, A 1 , A 2 , and A 3 w ere in jail. T hey knew that one of them w as to b e set free and the other t w o were to b e executed. Th ey did n ot kn o w 2 who was the one to b e spared, but the emp eror did know. A 1 said to the emp eror, “ I al ready kno w that at least one the other tw o prison er s will b e executed, so if you tell me the name of one who w ill b e executed, you wo n’t ha v e give n m e any information ab out m y own execution”. After some thinking, the emp eror said, “ A 3 will b e executed.” Thereup on A 1 felt happier b ecause his c hance had increased from 1 3(=Num[ { A 1 ,A 2 ,A 3 } ]) to 1 2(=Num[ { A 1 ,A 2 } ]) . This prisoner A 1 ’s happ iness ma y or ma y not b e reasonable? E A 1 A 2 A 3 ✲ ✲ “ A 3 will b e ex ecuted” (Emp eror) The pur p ose of this pap er is to clarify Problem 1 (Mon ty Hall problem) and Pr oblem 2 (three prisoners problem ) as follo ws . (A1) Problem 1 (Mon t y Hall pr ob lem) is sol v able, but Pr ob lem 2 (Th ree prisoners p roblem) is not w ell p osed. In this sense, P roblem 1 and Problem 2 are not equiv alen t. This is the direct consequence of Fisher’s maximal like liho o d metho d mentioned in Section 2. (A2) Also, there are t wo wa ys that the probabilistic pr op erty is introd uced to b oth problems as follo ws: (A2 1 ) in Problem 1, one (discussed in Section 4) is that the host casts the dice, and another (discussed in Section 6) is that yo u cast the dice. (A2 2 ) in Pr oblem 2, one (discussed in Section 4) is that th e emp eror casts th e dice, and another (discussed in Section 6) is that three prisons cast the dice. In the case of eac h, the former solution is du e to Bay es’ metho d ( men tioned in Section 2). And the latter solution is du e to the p rinciple is equal probabilities ( mentioned in Section 5). And, after all, we can conclude, under the situation (A2), that Prob lem 1 and Problem 2 are equiv alent. The ab o ve will b e sho wn in terms of quan tu m language (=measurement theory). And there- fore, w e exp ect the readers to find th at quantum language is su p erior to K olmogoro v’s probability theory [15]. 1.2 Ov erview: Measuremen t Theory (= Quan tum Language) As emphasized in refs. [6, 7], measur ement theory (or in short, MT) is, b y a linguistic turn of quan tum mec h anics (cf. Figure 1 : 7  later), constructed as the scien tific theory form ulated in a certain C ∗ -algebra A (i.e., a norm closed subalgebra in the op erator algebra B ( H ) comp osed of all b ounded op erators on a Hilb ert space H , cf. [16, 1 7] ). MT is comp osed of t w o th eories (i.e., pure measuremen t theory (or, in short, PMT] and statistical measurement th eory (or, in sh ort, SMT). T hat is, it has the follo win g structure: 3 (B) MT (measurement theory = quantum language) =          (B1) : [PMT ] = [(pure) measurement ] (Axiom P 1) + [causalit y] (Axiom 2) (B2) : [SMT ] = [(statistica l) measuremen t] (Axiom S 1) + [causalit y] (Axiom 2) where Axiom 2 is common in PMT and SMT. F or completeness, note th at measurement theory (B) (i.e., (B1) and (B2)) is not physics but a kind of language based on “the quan tum mechanica l w orld view”. As seen in [8], note that MT gives a found ation to stat istics. That is, roughly sp eaking, (C1) it ma y b e un derstandable to co nsider that PMT and SMT is related to Fisher’s statistics and Ba y esian statistics resp ectiv ely . When A = B c ( H ), the C ∗ -algebra co mp osed of all compact op erators on a Hilb ert space H , the (B) is called quantum measurement theory (or, quantum system th eory), which can b e regarded as the linguistic asp ect of quant um mec han ics. Also, when A is co mm utativ e  that is, when A is c h aracterized by C 0 (Ω), the C ∗ -algebra comp osed of all con tin u ous complex-v alued functions v anishing at infin it y on a lo cally compact Hausdorff s pace Ω (cf. [16])  , the (B) is called classical m easuremen t theory. Th u s, we h a ve the follo w in g classification: (C2) MT    quan tum MT (when A = B c ( H )) classical MT (when A = C 0 (Ω)) Also, f or th e p osition of MT in science, see Figure 1 , which w as precisely explained in [7, 10]. P armenides Socrates Greek philosophy Plato Alistotle Schola - − − − − − → sticism 1  − − → (monism) Newton (realism) 2  → relativit y theory − − − − − − → 3  → quan tum mec hanics − − − − − − → 4  − − → (dualism) Descartes Ro c k,... Kan t (idealism) 6  − − → (linguistic view) language philosophy quanti- − − − − − → zation 8  language − − − − − − → 7         5  − − → (unsolv ed) theory of ev erythin g (quan tum phys.)            10  − − → (=MT) quan tum language (language) Figure 1: The history of the w orld-view statistics system theory − − − − → 9  linguistic view realistic view 2 Classical Measuremen t Theory (Axioms and In terpretation) 2.1 Mathematical Preparations Since our concern is concen trated to the Mont y Hall problem and three prisoners p roblem, w e d ev ote our selv es to classical MT in (C2). 4 Throughout this pap er, w e assume th at Ω is a co mpact Hausdorff sp ace. T hus, w e can pu t C 0 (Ω) = C (Ω), which is defi ned b y a Banac h space (o r precisely , a co mm u tativ e C ∗ -algebra ) comp osed of all cont in uous complex-v alued fu nctions on a compact Hausdorff space Ω, where its norm k f k C (Ω) is defined b y max ω ∈ Ω | f ( ω ) | . Let C (Ω) ∗ b e the d ual Banac h space of C (Ω). That is, C (Ω) ∗ = { ρ | ρ is a con tin u - ous linear functional on C (Ω) } , and the norm k ρ k C (Ω) ∗ is defined by su p {| ρ ( f ) | : f ∈ C (Ω) su c h that k f k C (Ω) ≤ 1 } . The b i-linear f unctional ρ ( f ) is also denoted b y C (Ω) ∗ h ρ, f i C (Ω) , or in sh ort h ρ, f i . Defin e the mixe d state ρ ( ∈ C (Ω) ∗ ) suc h th at k ρ k C (Ω) ∗ = 1 an d ρ ( f ) ≥ 0 for all f ∈ C (Ω) suc h th at f ≥ 0. And p ut S m ( C (Ω ) ∗ )= { ρ ∈ C (Ω) ∗ | ρ is a mixed state } . Also, for eac h ω ∈ Ω, defin e the pur e state δ ω ( ∈ S m ( C (Ω ) ∗ )) suc h that C (Ω) ∗ h δ ω , f i C (Ω) = f ( ω ) ( ∀ f ∈ C (Ω)). And put S p ( C (Ω ) ∗ )= { δ ω ∈ C (Ω) ∗ | δ ω is a pu re state } , whic h is called a state sp ac e. Note, by the Riesz theorem (cf. [18] ), th at C (Ω) ∗ = M (Ω) ≡ { ρ | ρ is a signed measure on Ω } and S m ( C (Ω ) ∗ ) = M m +1 (Ω) ≡ { ρ | ρ is a measure o n Ω suc h that ρ (Ω) = 1 } . Also, it is clear that S p ( C (Ω ) ∗ ) = { δ ω 0 | δ ω 0 is a p oin t measure at ω 0 ∈ Ω } , wh ere R Ω f ( ω ) δ ω 0 ( dω ) = f ( ω 0 ) ( ∀ f ∈ C (Ω)). This imp lies that the state space S p ( C (Ω ) ∗ ) can b e also iden tified w ith Ω (called a sp e ctrum sp ac e or s im p ly , sp e ctrum ) s uc h as S p ( C (Ω ) ∗ ) (state space) ∋ δ ω ↔ ω ∈ Ω (spectrum) (1) Also, note that C (Ω) is u nital, i.e., it has the iden tit y I (or precisely , I C (Ω) ), since w e assume that Ω is compact. According to the noted idea (cf. [1]) in quantum mec h anics, an observable O :=( X , F , F ) in C (Ω) is d efined as follo ws: (D 1 ) [Field] X is a set, F ( ⊆ 2 X , the p o wer set o f X ) is a field of X , that is, “Ξ 1 , Ξ 2 ∈ F ⇒ Ξ 1 ∪ Ξ 2 ∈ F ”, “Ξ ∈ F ⇒ X \ Ξ ∈ F ”. (D 2 ) [Additivit y] F is a mapp ing from F to C (Ω) sati sfying: (a): for ev ery Ξ ∈ F , F (Ξ) is a non-negativ e element in C (Ω) such that 0 ≤ F (Ξ) ≤ I , (b): F ( ∅ ) = 0 and F ( X ) = I , wh ere 0 and I is the 0-elemen t and the identit y in C (Ω) resp ectiv ely . (c): for any Ξ 1 , Ξ 2 ∈ F suc h that Ξ 1 ∩ Ξ 2 = ∅ , it holds that F (Ξ 1 ∪ Ξ 2 ) = F (Ξ 1 ) + F (Ξ 2 ). F or th e more pr ecise argu m en t (su c h as counta bly additivit y , etc.), see [8]. 2.2 Classical PMT in (B1) In this s ection we shall explain classical PMT in (A 1 ). With any system S , a comm u tative C ∗ -algebra C (Ω) can b e asso ciated in whic h the m easure- men t theory ( B) of that sys tem can be form u lated. A state of the system S is repr esen ted by an elemen t δ ω ( ∈ S p ( C (Ω ) ∗ )) and an observable is represent ed by an observ able O :=( X, F , F ) in C (Ω). Also, the me asur ement of the observable O for the sys tem S with th e state δ ω is denoted b y M C (Ω) ( O , S [ δ ω ] )  or more pr ecisely , M C (Ω) ( O :=( X , F , F ) , S [ δ ω ] )  . An observer can obtain a measured v alue x ( ∈ X ) by the measuremen t M C (Ω) ( O , S [ δ ω ] ). 5 The Axiom P 1 present ed b elo w is a kind of mathematical generalizatio n of Born’s probabilistic in terpretation of qu an tu m mec hanics. And th us, it is a statement withou t realit y . Axiom P 1 [Classical PMT]. The pr ob ability that a me asur e d value x ( ∈ X ) obta ine d b y the me asur ement M C (Ω) ( O := ( X , F , F ) , S [ δ ω 0 ] ) b elongs to a set Ξ( ∈ F ) is given by [ F (Ξ)]( ω 0 ) . Next, w e explain Axiom 2 in (B). Let ( T , ≤ ) b e a tree, i.e., a partial ordered set suc h that “ t 1 ≤ t 3 and t 2 ≤ t 3 ” implies “ t 1 ≤ t 2 or t 2 ≤ t 1 ” . In this pap er, w e assume that T is finite. Assume that there exists an element t 0 ∈ T , called the r o ot of T , such that t 0 ≤ t ( ∀ t ∈ T ) holds. Put T 2 ≤ = { ( t 1 , t 2 ) ∈ T 2 | t 1 ≤ t 2 } . T h e family { Φ t 1 ,t 2 : C (Ω t 2 ) → C (Ω t 1 ) } ( t 1 ,t 2 ) ∈ T 2 ≤ is called a c ausal r elation ( d ue to the Heisenb er g pictur e ), if it sat isfies the follo wing conditions (E 1 ) and (E 2 ). (E 1 ) With eac h t ∈ T , a C ∗ -algebra C (Ω t ) is asso ciated. (E 2 ) F o r ev ery ( t 1 , t 2 ) ∈ T 2 ≤ , a Marko v op erator Φ t 1 ,t 2 : C (Ω t 2 ) → C (Ω t 1 ) is defin ed (i.e., Φ t 1 ,t 2 ≥ 0, Φ t 1 ,t 2 ( I C (Ω t 2 ) ) = I C (Ω t 1 ) ). An d it satisfies that Φ t 1 ,t 2 Φ t 2 ,t 3 = Φ t 1 ,t 3 holds for an y ( t 1 , t 2 ), ( t 2 , t 3 ) ∈ T 2 ≤ . The family of dual op er ators { Φ ∗ t 1 ,t 2 : S m ( C (Ω t 1 ) ∗ ) → S m ( C (Ω t 2 ) ∗ ) } ( t 1 ,t 2 ) ∈ T 2 ≤ is calle d a dual c ausal r elation ( due to the Schr¨ od inger pictur e ). When Φ ∗ t 1 ,t 2 ( S p ( C (Ω t 1 ) ∗ ) ⊆ S p ( C (Ω t 2 ) ∗ ) holds for an y ( t 1 , t 2 ) ∈ T 2 ≤ , the causal relation is said to b e d eterministic. Here, Axiom 2 in the measurement theory (B) is presen ted as follo ws: Axiom 2 [Causalit y]. The c ausality is r epr esente d by a c ausal r elation { Φ t 1 ,t 2 : C (Ω t 2 ) → C (Ω t 1 ) } ( t 1 ,t 2 ) ∈ T 2 ≤ . F or the fu r ther argumen t (i.e., the W ∗ -algebraic formulati on) of measurement theory , see App end ix in [6]. 2.3 Classical SMT in (B2) It is u sual to consider that w e do not kno w the s tate δ ω 0 when we tak e a measuremen t M C (Ω) ( O , S [ δ ω 0 ] ). That is b ecause w e usually tak e a measurement M C (Ω) ( O , S [ δ ω 0 ] ) in order to kno w the state δ ω 0 . Th us, when we w ant to emphasize that w e do not kno w the the stat e δ ω 0 , M C (Ω) ( O , S [ δ ω 0 ] ) is denoted b y M C (Ω) ( O , S [ ∗ ] ). Also, when w e kno w the distribu tion ν 0 ( ∈ M m +1 (Ω) = S m ( C (Ω ) ∗ )) of the u nkno wn state δ ω 0 , the M C (Ω) ( O , S [ δ ω 0 ] ) is d enoted b y M C (Ω) ( O , S [ ∗ ] ( ν 0 )). The Axiom S 1 presen ted b elo w is a kind of m athematical generalizati on of Axiom P 1. Axiom S 1 [C lassical SMT]. The pr ob ability that a me asur e d value x ( ∈ X ) obtaine d by the me asur ement M C (Ω) ( O :=( X, F , F ) , S [ ∗ ] ( ν 0 )) b elongs to a set Ξ( ∈ F ) i s given by ν 0 ( F (Ξ)) ( = C (Ω) ∗ h ν 0 , F (Ξ) i C (Ω) ) . R emark 1 . Note that tw o statistical measurements M C (Ω) ( O , S [ δ ω 1 ] ( ν 0 )) and M C (Ω) ( O , S [ δ ω 2 ] ( ν 0 )) can n ot b e distinguished b efore measurements. In this sense, we consider that, ev en if ω 1 6 = ω 2 , w e can assu me th at M C (Ω) ( O , S [ δ ω 1 ] ( ν 0 )) = M C (Ω) ( O , S [ ∗ ] ( ν 0 )) = M C (Ω) ( O , S [ δ ω 2 ] ( ν 0 )) . (2) 2.4 Linguistic In t erpretation Next, w e ha v e to answ er ho w to u se the ab o ve axioms as follo ws. That is, we present the follo wing linguistic in terpretation (F) [=(F 1 )–(F 3 )], whic h is c haracterized as a kind of linguistic turn of so-calle d C op enhagen in terpr etation (cf. [6, 7] ). 6 That is, we p rop ose: (F 1 ) Consider the du alism comp osed of “observer” and “system( =measurin g ob ject)” suc h as • observer (I(=mind)) system (matter) ✛ ✲ observable measured value a  inte rfere b  p erceiv e a reaction state Figure 2. Descartes’ figure in MT And therefore, “observ er” and “system” must b e absolutely separated. (F 2 ) Only one measuremen t is p er m itted. And th us, the state after a measur emen t is meaningless since it can not b e measured any longer. Also, the causalit y should b e assumed only in the s ide of system, ho wev er, a state never mo v es. T h us, the Heisenberg picture should b e adopted. (F 3 ) Also, the observe r do es not h a ve the space-time. T hus, the question: “When and where is a measured v alue obtained?” is out of measur ement theory , and so on. This inte rpretation is, of course, common to b oth PMT and SMT. 2.5 Preliminary F undamen tal Theorems W e h a ve the follo w ing t w o f undamental theorems in m easur emen t th eory: Theorem 1 [Fisher’s maximum lik eliho o d metho d (cf. [8])]. Assume that a measured v alue obtained b y a mea surement M C (Ω) ( O := ( X , F , F ) , S [ ∗ ] ) b elongs to Ξ ( ∈ F ). Then, there is a reason to infer th at the un kno w n state [ ∗ ] is equal to δ ω 0 , where ω 0 ( ∈ Ω ) is defin ed by [ F (Ξ)]( ω 0 ) = max ω ∈ Ω [ F (Ξ)]( ω ) . Theorem 2 [Ba y es’ method (c f. [8])]. Assume that a m easured v alue obtained b y a statistica l measuremen t M C (Ω) ( O := ( X , F , F ) , S [ ∗ ] ( ν 0 )) b elongs to Ξ ( ∈ F ). Th en, there is a reason to infer that the p osterior state (i.e., the mixed state after the measurement ) is equal to ν post , which is defined b y ν post ( D ) = R D [ F (Ξ)]( ω ) ν 0 ( dω ) R Ω [ F (Ξ)]( ω ) ν 0 ( dω ) ( ∀ D ∈ B Ω ; Borel field) . The ab ov e t w o theorems are, of course, the most fun damen tal in statistics. Thus, we b eliev e in Figure 1 , i.e., statistics − − − − − − − → 9  10  quan tum language 7 3 T he First Answ er to Mon t y Hall P roblem [resp. Three pris- oners problem] b y F isher’s metho d In this section, w e pr esent the fi rst answer to Problem 1 (Mon ty-Ha ll p roblem) [resp. Problem 2 (Three p r isoners problem)] in classical PMT. The tw o w ill b e sim u ltaneously solve d as follo ws. The spirit of d ualism (in Figure 2) urges us to d eclare th at (G) ”observer ≈ y ou” and ”system ≈ three do ors” in Problem 1 [resp. ”observ er ≈ p risoner A 1 ” and ”system ≈ emp eror’s mind” in Pr oblem 2] Put Ω = { ω 1 , ω 2 , ω 3 } with th e discrete to p ology . Assume th at eac h state δ ω m ( ∈ S p ( C (Ω ) ∗ )) means δ ω m ⇔ the state that the car is b ehind the do or A m [resp. δ ω m ⇔ the state that the prisoner A m will b e f r ee ] ( m = 1 , 2 , 3) (3) Define the obs er v able O 1 ≡ ( { 1 , 2 , 3 } , 2 { 1 , 2 , 3 } , F 1 ) in C (Ω ) such that [ F 1 ( { 1 } )] ( ω 1 ) = 0 . 0 , [ F 1 ( { 2 } )] ( ω 1 ) = 0 . 5 , [ F 1 ( { 3 } )] ( ω 1 ) = 0 . 5 , [ F 1 ( { 1 } )] ( ω 2 ) = 0 . 0 , [ F 1 ( { 2 } )] ( ω 2 ) = 0 . 0 , [ F 1 ( { 3 } )] ( ω 2 ) = 1 . 0 , [ F 1 ( { 1 } )] ( ω 3 ) = 0 . 0 , [ F 1 ( { 2 } )] ( ω 3 ) = 1 . 0 , [ F 1 ( { 3 } )] ( ω 3 ) = 0 . 0 , (4) where it is also p ossible to assu me that F 1 ( { 2 } )( ω 1 ) = α , F 1 ( { 3 } )( ω 1 ) = 1 − α (0 < α < 1). Th us w e h a ve a measuremen t M C (Ω) ( O 1 , S [ ∗ ] ), whic h should b e regarded as the measuremen t theoretical rep r esen tatio n of the measurement th at you say ”Do or A 1 ” [resp. ”P risoner A 1 ” asks to the emp er or ]. Here, w e assum e that a) “measured v alue 1 is obtained by the measur emen t M C (Ω) ( O 1 , S [ ∗ ] )” ⇔ Th e h ost sa ys “Do or A 1 has a goat” [resp. ⇔ th e emp eror sa ys “Prisoner A 1 will b e executed” ] b) “measured v alue 2 is obtained by the measur emen t M C (Ω) ( O 1 , S [ ∗ ] ) ” ⇔ Th e h ost sa ys “Do or A 2 has a goat” [resp. ⇔ th e emp eror sa ys “Prisoner A 2 will b e executed” ] c) “measured v alue 3 is obtained b y th e measur ement M C (Ω) ( O 1 , S [ ∗ ] ) ” ⇔ Th e h ost sa ys “Do or A 3 has a goat” [resp. ⇔ th e emp eror sa ys “Prisoner A 3 will b e executed” ] Recall that, in Pr oblem 1 (Mont y-Hall prob lem) [resp. Pr oblem 2 (Three prisoners pr oblem)] , the host said “Door 3 has a goat” [resp. the emp eror s aid “Prisoner A 3 wil b e executed”] T his implies that yo u [resp. “Prisoner A 1 ] get the measured v alue “3” by the measurement M C (Ω) ( O 1 , S [ ∗ ] ). Note that [ F 1 ( { 3 } )] ( ω 2 ) = 1 . 0 = max { 0 . 5 , 1 . 0 , 0 . 0 } = m ax { [ F 1 ( { 3 } )] ( ω 1 ) , [ F 1 ( { 3 } )] ( ω 2 ) , [ F 1 ( { 3 } )] ( ω 3 ) } , (5) Therefore, Theorem 1 (Fisher’s maximum like liho o d metho d ) sa ys that (H1) In Prob lem 1 (Mont y-Hall problem), there is a reaso n to infer that [ ∗ ] = δ ω 2 . Thus, yo u should switc h to Do or A 2 . 8 (H2) In Problem 2 (Three prisoners problem), th ere is a reason to infer that [ ∗ ] = δ ω 2 . Ho w ever, there is no reasonable ans w er for the question: whether Prisoner A 1 ’s happ iness increases. That is, P r oblem 2 is n ot a we ll-p osed problem. 4 The Second Answe r to M on t y Hall Problem [resp. Three prisoners problem] b y B a y es’ metho d In o rder to use Ba y es’ metho d, shall mo dify Pr oblem 1(Mon t y Hall problem) and Problem 2(Th r ee prisoners problem) as follo ws. 4.1 Problems 1 ′ and 2 ′ ( Mon ty Hall P roblem [resp. Three prisoners problem] ) Problem 1 ′ [Mon ty Hall problem; the host casts the dice]. Supp ose you are on a game sho w, and y ou are give n the c h oice of three d o ors (i.e., “Door A 1 ” , “Do or A 2 ” , “Do or A 3 ” ). Behind one do or is a car, b ehind the others, goats. Y ou do not kno w w hat’s b ehind the do ors. Ho wev er, y ou pick a do or, sa y ”Door A 1 ”, and the host, who knows w hat’s b ehind the doors, op ens another do or, s ay “Do or A 3 ” , wh ich has a goat. And he adds that ( ♯ 1 ) the c ar was set b ehind the do or de cide d b y the c ast of the (distorte d) dic e. That is, the host set the c ar b ehind Do or A m with pr ob ability p m (wher e p 1 + p 2 + p 3 = 1 , 0 ≤ p 1 , p 2 , p 3 ≤ 1 ) . He sa ys to y ou, “Do y ou wa n t to pic k Do or A 2 ?” Is it to y our adv an tage to switc h yo ur c hoice of do ors? ❄ ❄ ❄ Do or A 1 Do or A 2 Do or A 3 Problem 2 ′ [Three p risoners pr oblem; the emp eror casts th e dice]. Three prisoners, A 1 , A 2 , and A 3 w ere in jail. They knew that one of t hem w as to b e set free and the other t wo were to b e executed. They did not know who w as the one to b e spared, but th ey kno w that ( ♯ 2 ) the one t o b e sp ar e d was de cide d by the c ast of the (disto rte d) dic e. That is, Prisoner A m is to b e sp ar e d with pr ob ability p m (wher e p 1 + p 2 + p 3 = 1 , 0 ≤ p 1 , p 2 , p 3 ≤ 1 ) . but the emp eror did kno w the one to b e spared. A 1 said to the emp eror, “I already know that at lea st one the other t wo prisoners will b e executed, so if y ou tell me the name of one who will b e exe cuted, y ou w on ’t h a ve giv en me an y information ab out m y o wn execution”. Afte r some thinking, the emp eror said, “ A 3 will b e executed.” Thereup on A 1 felt happier b ecause his c hance had increased from 1 3(=Num[ { A 1 ,A 2 ,A 3 } ]) to 1 2(=Num[ { A 1 ,A 2 } ]) . This p risoner A 1 ’s happ iness ma y or ma y not b e reasonable? 9 E A 1 A 2 A 3 ✲ ✲ “ A 3 will b e ex ecuted” (Emp eror) R emark 2 . In Problem 1 ′ , y ou ma y choose ”D o or A 1 ” by v arious w a ys. F or e xample, y ou may c ho ose ”Do or A 1 ” b y the metho d mentio ned in Problem 1 ′′ later. 4.2 The second answ er t o Problems 1 ′ and 2 ′ ( Mon t y Hall P roblem [resp. Three prisoners problem] ) b y Ba y es’ metho d In what follo ws we stud y th ese problems. Let Ω and O 1 b e as in Section 3 . Und er the hyp othesis ( ♯ 1 ) [resp. ( ♯ 2 )] , define the mixed state ν 0 ( ∈ M m +1 (Ω)) suc h that: ν 0 ( { ω 1 } ) = p 1 , ν 0 ( { ω 2 } ) = p 2 , ν 0 ( { ω 3 } ) = p 3 (6) Th us we hav e a statistica l measuremen t M C (Ω) ( O 1 , S [ ∗ ] ( ν 0 )). Note that a) “measured v alue 1 is obtained by the statistical measurement M C (Ω) ( O 1 , S [ ∗ ] ( ν 0 ))” ⇔ the h ost says “Do or A 1 has a goat” [resp. ⇔ th e emp eror sa ys “Prisoner A 1 will b e executed” ] b) “measured v alue 2 is obtained by the statistical measurement M C (Ω) ( O 1 , S [ ∗ ] ( ν 0 ))” ⇔ the h ost says “Do or A 2 has a goat” [resp. ⇔ th e emp eror sa ys “Prisoner A 2 will b e executed” ] c) “measured v alue 3 is obtained b y th e statistical measurement M C (Ω) ( O 1 , S [ ∗ ] ( ν 0 ))” ⇔ th e host sa ys “Do or A 3 has a goat” [resp. ⇔ th e emp eror sa ys “Prisoner A 3 will b e executed” ] Here, assu m e that, by the statistical measuremen t M C (Ω) ( O 1 , S [ ∗ ] ( ν 0 )), y ou obtain a measured v alue 3, which corresp ond s to th e fact that the host said “Do or A 3 has a goat” . [resp. the emp eror said that Prisoner A 3 is to b e executed ], Then, T heorem 2 (Ba yes’ metho d) s a ys that the p osterior state ν post ( ∈ M m +1 (Ω)) is giv en by ν post = F 1 ( { 3 } ) × ν 0  ν 0 , F 1 ( { 3 } )  . (7) That is, ν post ( { ω 1 } ) = p 1 2 p 1 2 + p 2 , ν post ( { ω 2 } ) = p 2 p 1 2 + p 2 , ν post ( { ω 3 } ) = 0 . (8) Then, 10 (I1) In Pr ob lem 1 ′ ,    if ν post ( { ω 1 } ) < ν post ( { ω 2 } ) (i.e., p 1 < 2 p 2 ), y ou sh ould pick Do or A 2 if ν post ( { ω 1 } ) = ν post ( { ω 2 } ) (i.e., p 1 < 2 p 2 ), y ou may pick Do ors A 1 or A 2 if ν post ( { ω 1 } ) > ν post ( { ω 2 } ) (i.e., p 1 < 2 p 2 ), y ou sh ould not pick Do or A 2 (I2) In Pr ob lem 2 ′ ,    if ν 0 ( { ω 1 } ) < ν post ( { ω 1 } ) (i.e., p 1 < 1 − 2 p 2 ), the pr isoner A 1 ’s happiness increases if ν 0 ( { ω 1 } ) = ν post ( { ω 1 } ) (i.e., p 1 = 1 − 2 p 2 ), the pr isoner A 1 ’s happiness is inv ariant if ν 0 ( { ω 1 } ) > ν post ( { ω 1 } ) (i.e., p 1 > 1 − 2 p 2 ), the pr isoner A 1 ’s happiness decreases 5 The Principle of Equal Probabilit y In this section, according to [4, 6, 11] w e prepare Th eorem 3 (the principle of equal probabilit y), i.e., (J) unless w e ha ve sufficient reason to regard on e p ossible case as more probab le than another, w e tr eat them as equ ally pr obable. This theorem will b e u sed in the follo win g section. Put Ω = { ω 1 , ω 2 , ω 3 , . . . , ω n } with the discrete top ology . And consider any observ able O 1 ≡ ( X, F , F 1 ) in C (Ω ). Define the b ijection φ 1 : Ω → Ω such that φ 1 ( ω j ) =  ω j +1 ( j 6 = n ) ω 1 ( j = n ) and define the observ able O k ≡ ( X , F , F k ) in C (Ω ) such that [ F k (Ξ)]( ω ) = [ F 1 (Ξ)]( φ k − 1 ( ω )) ( ∀ ω ∈ Ω , k = 1 , 2 , ..., n ) where φ 0 ( ω ) = ω ( ∀ ∈ Ω) and φ k ( ω ) = φ 1 ( φ k − 1 ( ω )) ( ∀ ω ∈ Ω , k = 1 , 2 , ..., n ). Let p k ( k = 1 , ..., n ) b e a n on-negativ e real num b er such that P n k =1 p k = 1. (K) F or example, fix a state δ ω m ( m = 1 , 2 , ..., n ). And, b y the cast of th e ( distorted ) dice, y ou c ho ose an observ able O k ≡ ( X , F , F k ) with probability p k . An d further, yo u take a measuremen t M C (Ω) ( O k := ( X , F , F k ) , S [ δ ω m ] ). Here, we can easily see that th e probabilit y th at a measur ed v alue obtained b y the measur emen t (K) b elongs to Ξ ( ∈ F ) is given b y n X k =1 p k h F k (Ξ) , δ ω m i  = n X k =1 p k [ F k (Ξ)]( ω m )  (9) 11 whic h is equal to h F 1 (Ξ) , P n k =1 p k δ φ k − 1 ( ω m ) i . This implies th at the measurement (K) is equiv alen t to a statistica l measuremen t M C (Ω) ( O 1 := ( X, F , F 1 ) , S [ δ ω m ] ( P n k =1 p k δ φ k − 1 ( ω m ) )). Note that the (9) d ep ends on th e state δ m . Th us, we can not calcula te the (9) such as th e (8). Ho wev er, if it holds that p k = 1 /n ( k = 1 , ..., n ), w e see that 1 n P n k =1 δ φ k − 1 ( ω m ) is indep endent of the choice of the state δ ω m . Thus, pu tting 1 n P n k =1 δ φ k − 1 ( ω m ) = ν e , w e see that th e measurement (K) is equiv alen t to the statistica l measuremen t M C (Ω) ( O 1 , S [ δ ω m ] ( ν e )), wh ic h is also equiv alen t to M C (Ω) ( O 1 , S [ ∗ ] ( ν e )) (from the formula (2) in Remark 1). Th us, un der the ab o ve n otation, w e hav e the f ollo wing theorem, whic h realizes the s p irit (J). Theorem 3 [ The prin ciple of equ al probability (i.e., the equal probabilit y of selection) ]. If p k = 1 /n ( k = 1 , ..., n ), the m easuremen t (K) is indep end en t of the c hoice of the state δ m . He nce, the (K) is equiv alent to a statistica l measuremen t M C (Ω) ( O 1 := ( X , F , F 1 ) , S [ ∗ ] ( ν e )). It should b e noted that the prin ciple of equ al probabilit y is not ”principle” but ”theorem” in measuremen t theory . R emark 3 . In the ab o ve argument, w e consid er the set B ′ = { φ k | k = 1 , 2 , ..., n } . Ho wev er, it ma y b e m ore n atur al to consider the set B = { φ | φ : Ω → Ω is a bijection } . 6 The Third Answ er to Mon t y Hall Problem [resp. Three pris- oners problem] b y the principle of equal probabilit y 6.1 Problems 1 ′′ and 2 ′′ ( Mont y Hall P roblem [resp. Three prisoners problem] ) Problem 1 ′′ [Mon ty Hall problem; you cast the d ice]. Supp ose yo u are on a game show, and y ou are giv en the c hoice of three do ors (i.e., “Do or A 1 ” , “Do or A 2 ” , “Do or A 3 ” ). Behind one d o or is a car, b ehind the others, goats. Y ou do not kno w w hat’s b ehind the do ors. Th u s, ( ♯ 1 ) you sele ct D o or A 1 by the c ast of the fair dic e. That is, you say ”Do or A 1 ” with pr ob ability 1/3. The host, who kno ws what’s b ehind the d o ors, op ens another do or, s ay “Door A 3 ” , whic h has a goat. He says to you, “Do yo u wan t to p ic k Do or A 2 ?” I s it to your adv an tage to switc h y our c hoice of d o ors? ❄ ❄ ❄ Do or A 1 Do or A 2 Do or A 3 Problem 2 ′′ [Three p risoners problem; the pr isoners cast the dice]. Three p risoners, A 1 , A 2 , and A 3 w ere in jail. They knew that one of t hem w as to b e set free and the other t wo were to b e executed. They d id n ot kno w who was the one to be spared, b ut the emp eror did kno w. S ince three prisoners w anted to ask the emp eror, ( ♯ 2 ) the questioner was de cide d by the fair die thr ow. And Prisoner A 1 was sele cte d with pr ob a- bility 1 / 3 12 Then, A 1 said to th e emp eror, “I already kn o w that at least one the other tw o p risoners will b e executed, so if y ou tell me the name of one who will b e executed, y ou w on’t ha ve giv en me an y information ab out m y o wn execution”. After s ome thin king, the emp eror said, “ A 3 will b e executed.” Thereup on A 1 felt happier b ecause his c hance h ad in creased from 1 3(=Num[ { A 1 ,A 2 ,A 3 } ]) to 1 2(=Num[ { A 1 ,A 2 } ]) . This pr isoner A 1 ’s h appiness m a y or ma y not b e reasonable? E A 1 A 2 A 3 ✲ ✲ “ A 3 will b e ex ecuted” (Emp eror) Answ er: By Th eorem 3 (Th e p rinciple of equal probabilit y), the ab ov e Problems 1 ′′ and 2 ′′ is resp ectiv ely the same as Pr oblems 1 ′ and 2 ′ in the case that p 1 = p 2 = p 3 = 1 / 3. Then, the form ulas (6) and (8) sa y that (L1) In Pr ob lem 1 ′′ , since ν post ( { ω 1 } ) = 1 / 3 < 2 / 3 = ν post ( { ω 2 } ), you s h ould p ick Do or A 2 . (L2) In Pr oblem 2 ′′ , since ν 0 ( { ω 1 } ) = 1 / 3 = ν post ( { ω 1 } ), the prisoner A 1 ’s happiness is inv ariant. 7 Conclusions Although main idea is due to r efs. [5, 11], in this pap er we simultaneously discussed the Mon t y Hall problem and the th r ee prisoners problem in terms of qu antum language. Th at is, we gav e three answ ers, i.e., (M1) the fi rst answer (d ue to Fisher’s metho d) in Section 3, (M2) the s econd answer (du e to Ba yes’ metho d) in Section 4, (M3) the th ird answer (due to Theorem 3(the principle of equal probabilit y)) in Section 6 W e of course b eliev e that ou r pr op osal is the fin al solutions of the t w o problems. 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