9.9: Fisher statistics: Monty Hall problem [three prisoners problem]

But, we think that the meaning of "isomorphism problem" is not clarified, or, it is not able to be clarified without measurement (or, the dualism).

Problem 9.21(=Problem9.15Problem [Monty Hall problem])

$\quad$

Suppose you are on a game show, and you are given the choice of three doors (i.e., " Door $A_1$"$\!\!\!,\;$ " Door $A_2$"$\!\!\!,\;$ " Door $A_3$"$\!\!)$. Behind one door is a car, behind the others, goats. You do not know what's behind the doors However, you pick a door, say "Door $A_1$", and the host, who knows what's behind the doors, opens another door, say " Door $A_3$"$\!\!\!,\;$ which has a goat. He says to you, " Do you want to pick Door $A_2$?"$\;\;$ Is it to your advantage to switch your choice of doors?

Problem 9.22[three prisoners problem]

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Three prisoners, $A_1$, $A_2$, and $A_3$ were in jail. They knew that one of them was to be set free and the other two were to be executed. They did not know who was the one to be spared, but the emperor did know. $A_1$ said to the emperor, "I already know that at least one the other two prisoners will be executed, so if you tell me the name of one who will be executed, you won't have given me any information about my own execution".$\;\;$ After some thinking, the emperor said, "$A_3$ will be executed."$\;\;$ Thereupon $A_1$ felt happier because his chance had increased from $\frac{1}{3(= {\rm Num}\{A_1,A_2,A_3 \}])}$ to $\frac{1}{2(= {\rm Num}\{ A_1,A_2 \}])}$. This prisoner $A_1$'s happiness may or may not be reasonable?

Descartes Figure 9.7: The image of "measurement" in dualism

In the dualism, we have the confrontation

\begin{align} \mbox{ "observer$\longleftrightarrow$system" } \end{align} as follows. Correspondence: observer$\cdot$system

In what follows, we present the first answer to $ \left[\begin{array}{l} \mbox{Problem 9.21 (Monty-Hall problem)} \\ \color{blue}{\mbox{Problem 9.22(Three prisoners problem)}} \end{array}\right] $ in classical pure measurement theory. The two will be simultaneously solved as follows. The spirit of dualism (in Figure 9.7) urges us to declare that

Put $\Omega = \{ \omega_{1} , \omega_{2} , \omega_{3} \}$ with the discrete topology. Assume that each state $\delta_{\omega_{{m}}} (\in {\frak S}^p (C(\Omega)^* ))$ means

Define the observable ${\mathsf O}_1$ $\equiv$ $(\{ 1, 2,3 \}, 2^{\{1, 2 ,3\}}, F_1)$ in $L^\infty (\Omega)$ such that

where it is also possible to assume that $F_1(\{ 2 \})(\omega_1)=\alpha$, $F_1(\{ 3 \})(\omega_1) =1- \alpha$ $ (0 < \alpha < 1)$. Thus we have a measurement ${\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_1, S_{[{}\ast{}]})$, which should be regarded as the measurement theoretical representation of the measurement that $ \left[\begin{array}{ll} \mbox{ you say "Door $A_1$" } \\ \color{blue}{ \mbox{ "Prisoner $A_1$" asks to the emperor} } \end{array}\right] $.

Recall that $ \left[\begin{array}{ll} \mbox{ the host said "Door 3 has a goat" } \\ \color{blue}{\mbox{ the emperor said "Prisoner $A_3$ will be executed" } } \end{array}\right] $.

This implies that $ \left[\begin{array}{ll} \mbox{you } \\ \color{blue}{\mbox{ Prisoner $A_1$ } } \end{array}\right] $ get the measured value "3" by the measurement ${\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_1,$ $ S_{[\ast]})$. Note that

Therefore, Theorem 5.6 (Fisher's maximum likelihood method) says that