9.11: Equal probability}: Monty Hall problem [three prisoners problem]

This section is extracted from the following:

 $(\sharp):$ S. Ishikawa; The Final Solutions of Monty Hall Problem and Three Prisoners Problem S. Ishikawa, "A Measurement Theoretical Foundation of Statistics," arXiv:1408.0963v1 [stat.OT] 2014

Problem 9.25 [(=Problem9.16) Monty Hall problem(the casethat you throws the dice)]
$\quad$ 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. Thus,
 $(\sharp_1):$ you select Door $A_1$ by the cast of the fair dice. That is, you say "Door $A_1$" with probability 1/3.

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 9.26[three prisoners problem( the case that the prisoner throws the dice)]
$\quad$ 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. Since three prisoners wanted to ask the emperor,
 $(\sharp_2):$ the questioner was decided by the fair die throw. And Prisoner $A_1$ was selected with probability $1/3$
Then, $A_1$ 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? By Theorem 9.18 (The principle of equal weight), the above Problems 9.25 and 9.26 is respectively the same as Problems 9.23 and 9.24 in the case that $p_1=p_2=p_3=1/3$.
 $(B_1):$ Problem9.25 [Monty Hall problem ( the case that you throw a fair dice)] \begin{align} & \mbox{ $\nu_{ post} (\{ \omega_1 \})$ $<$ $\nu_{ post} (\{ \omega_2 \})$ (i.e., $p_1=1/3 < 2/3=2 p_2$),} \\ & \mbox{ thus, you should choose the door $A_2$} \end{align} $(B_2):$ Problem9.26[three prisoners problem ( the case that the emperor throws a fair dice)], \begin{align} & \mbox{ $\nu_{0} (\{\omega_1\}) = \nu_{\rm post} (\{\omega_1\})$ (i.e., $p_1 =1/3= 1- 2 p_2$),} \\ & \mbox{ Thus, the happiness of the prisoner $A_1$ is invariant} \end{align}
 $\fbox{Note 9.6}$ These problems (i.e., Monty Hall problem and the three prisoners problem) continued attracting the philosopher's interest. This is not due to that these are easy to make a mistake for high school students, but these problems include the essence of "dualism".