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# 数学代写|信息论代写Information Theory代考|ECE1502 The Three Prisoners’ Problem

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## 数学代写|信息论代写Information Theory代考|The Three Prisoners’ Problem

This problem is a problem in probability theory. However, in this book, we shall view it as a problem within information theory.

Three prisoners, called A, B, and C, were sentenced to death. A few days before the scheduled execution date, the king decides that he wants to grant freedom to one of the prisoners. The king does not know the prisoners personally and therefore he also does not have any preferred one to be freed. Therefore, he decides to throw a die, the result of which will determine who shall be freed. He makes a special die on the faces of which are inscribed the letters A, A, B, B, C, C. The dice is fair, meaning that the probability of each of the letters, A, B and C, to appear is equal to $1 / 3$. Once the die is thrown and the result is known, the king orders the warden to free the prisoner bearing the name that came up on the face of the die, on the same day that the other two will be executed. He also orders the warden not to reveal to anyone the name of the lucky prisoners until the actual day of the execution.

All the details of the story about the king throwing the die are told to the prisoners.

Prisoner A is pondering his chances of survival. One day before the scheduled execution, he approaches the warden and says: “I know you that you know the results of throwing the die. I also know that you are not allowed to tell me who will be freed tomorrow However, I know for certainty that two of the three prisoners must be executed tomorrow. This means that either $\mathrm{B}$ or $\mathrm{C}$ will die tomorrow. All I am asking you is that you will tell me which one of these two (B or C) will be executed?
The warren thinks for a while. He concludes that by revealing to prisoner, named A, the identity of one prisoner who will be executed, he does not reveal the name of the one who will be freed. Hence, by answering prisoner A’s question, the warren will not be defying the king’s order. Therefore, the warden tells prisoner A that prisoner B is going to be executed the next day.

## 数学代写|信息论代写Information Theory代考|Two Solutions to the Three Prisoners’ Problem

We start by defining the following three events, and their probabilities:
\begin{aligned} A f & ={\text { Prisoner A is going to be freed }} \ \operatorname{Pr}(A f) & =\frac{1}{3} \ B f & ={\text { Prisoner B is going to be freed }} \ \operatorname{Pr}(B f) & =\frac{1}{3} \end{aligned}
$C f={$ Prisoner $\mathrm{C}$ is going to be freed $}$

$$\operatorname{Pr}(C f)=\frac{1}{3}$$
We also define the following three events, and their probabilities:
\begin{aligned} A d & ={\text { Prisoner A is going to die }} \ \operatorname{Pr}(A d) & =\frac{2}{3} \ B d & ={\text { Prisoner B is going to die }} \ \operatorname{Pr}(B d) & =\frac{2}{3} \ C d & ={\text { Prisoner } \mathrm{C} \text { is going to die }} \ \operatorname{Pr}(C d) & =\frac{2}{3} \end{aligned}

## 数学代写|信息论代写Information Theory代考|The Three Prisoners’ Problem

A、B、C三名囚犯被判处死刑。在预定的处决日期前几天，国王决定要给予其中一名囚犯自由。国王本人并不 认识这些囚犯，因此他也没有布望释放的人。因此，他决定郑骰子，结果将决定谁将被释放。他做了一个特殊 的骰子，上面刻着字母 $A 、 A 、 B 、 B 、 C 、 C$ 。骰子是公平的，这意味着字母 $A 、 B$ 和 C 中的每一个出现的概率 是等于 $1 / 3$. 一旦掷出骰子并知道结果，国王便命令典狱长在其他两人被处决的同一天释放骰子上名字出现的囚 犯。他还命令典狱长在执行死刑前不得向任何人透露幸运囚犯的姓名。

## 数学代写|信息论代写Information Theory代考|Two Solutions to the Three Prisoners’ Problem

$A f=$ Prisoner $\mathrm{A}$ is going to be freed $\operatorname{Pr}(A f) \quad=\frac{1}{3} B f=$ Prisoner $\mathrm{B}$ is going to be freed $\operatorname{Pr}(B f) \quad=\frac{1}{3}$
$C f=\$$Prisoner \ \mathrm{C} \$$ isgoingtobefreed$\
\operatorname{Pr}(C f)=\frac{1}{3}


$A d=$ Prisoner $\mathrm{A}$ is going to die $\operatorname{Pr}(A d) \quad=\frac{2}{3} B d=$ Prisoner $\mathrm{B}$ is going to die $\operatorname{Pr}(B d) \quad=\frac{2}{3} C d=$ Prisoner $\mathrm{C}$ is goine

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。