├── LaTeX Aesthetic Template.zip ├── main.tex ├── README.md ├── condprobexdice.tex ├── preamble.tex ├── lecture2.tex ├── condprobsimp.tex └── condprobvis.tex /LaTeX Aesthetic Template.zip: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/THS119/LaTeXAestheticTemplate/HEAD/LaTeX Aesthetic Template.zip -------------------------------------------------------------------------------- /main.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,a4paper]{book} 2 | 3 | \input{preamble} 4 | 5 | \begin{document} 6 | 7 | \input{lecture2} 8 | 9 | \end{document} 10 | 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # LaTeXAestheticTemplate 2 | This LaTeX template is designed to provide a clean, elegant, and highly readable document style, similar to the renowned Tufte class templates. It is particularly well-suited for academic writing, lab reports, assignments and perhaps thesis (will work on it in the future) and other documents where clarity helps the reader. 3 | 4 | The design of this template is inspired by the beautiful works of **Aditya Dhumuntarao** (https://github.com/Adhumunt/NotesTeX) and the late **Gilles Castell (RIP)** (https://castel.dev/), who have crafted visually appealing and functionally efficient LaTeX templates that blend simplicity with sophistication. This template takes inspiration from their approaches but is mostly based on personal touches. 5 | 6 | I intend to use this template in the future to create notes that cover undergraduate and post-graduate level courses in mathematics, computer science, and engineering. 7 | 8 | ## V1 9 | This version has been posted, and is still undergoing some minor editing, as I am utilizing this template to typeset my probability notes (to be posted on a separate repo soon). I would like to thank everyone in the LaTeX reddit community who offered some great advices to modify my old templates. 10 | 11 | Run the main.tex file, run in LuaLaTeX. 12 | 13 | ![RDT_20250607_0905296227409126416925867](https://github.com/user-attachments/assets/a45e51f2-cde6-484b-83e4-0f7da0679cea) 14 | 15 | -------------------------------------------------------------------------------- /condprobexdice.tex: -------------------------------------------------------------------------------- 1 | \pgfplotsset{compat = 1.16} 2 | 3 | \begin{tikzpicture}[scale=0.6,every node/.style={scale=2}] 4 | \begin{axis}[ 5 | colormap = {custom}{color(0)=(nblue!20) color(1)=(nblue!30) color(2)=(nblue!40) color(3)=(nblue!50) color(4)=(nblue!60) color(5)=(nblue!70) color(6)=(nblue!80) color(7)=(nblue!90) color(8)=(nblue) 6 | }, 7 | xlabel={$X$}, 8 | ylabel={$Y$}, 9 | xtick={1,2,3,4}, 10 | ytick={1,2,3,4}, 11 | xmin=0.5, xmax=4.5, 12 | ymin=0.5, ymax=4.5, 13 | axis on top, 14 | point meta min=2, 15 | point meta max=8, 16 | nodes near coords={\pgfmathprintnumber\pgfplotspointmeta}, 17 | nodes near coords align={center}, 18 | ] 19 | \addplot [ 20 | matrix plot*, 21 | mesh/cols=4, 22 | point meta=explicit, 23 | ] table [meta=C] { 24 | x y C 25 | 1 1 2 26 | 1 2 3 27 | 1 3 4 28 | 1 4 5 29 | 2 1 3 30 | 2 2 4 31 | 2 3 5 32 | 2 4 6 33 | 3 1 4 34 | 3 2 5 35 | 3 3 6 36 | 3 4 7 37 | 4 1 5 38 | 4 2 6 39 | 4 3 7 40 | 4 4 8 41 | }; 42 | 43 | \fill[red,draw=black,thick] (1.5,1.5) rectangle ++(1.5,0.5); 44 | \fill[yellow,draw=black,thick] (1.5,2.5) rectangle ++(1.5,0.5); 45 | \fill[red,draw=black,thick] (1.5,3.5) rectangle ++(1.5,0.5); 46 | \fill[yellow,draw=black,thick] (2.5,1.5) rectangle ++(1.5,0.5); 47 | \fill[yellow,draw=black,thick] (2.5,1.5) rectangle ++(1.5,0.5); 48 | \fill[red,draw=black,thick] (3.5,1.5) rectangle ++(1.5,0.5); 49 | \end{axis} 50 | \end{tikzpicture} 51 | -------------------------------------------------------------------------------- /preamble.tex: -------------------------------------------------------------------------------- 1 | \usepackage[utf8]{inputenc} 2 | \usepackage[T1]{fontenc} 3 | \usepackage{fourier} 4 | \usepackage{marginnote,sidenotes} 5 | \usepackage{cabin} 6 | \usepackage[nomarginpar]{geometry} 7 | \usepackage{etoolbox} 8 | \usepackage{microtype} 9 | \usepackage{lettrine} 10 | \RequirePackage{marginfix}% automatically adjust the side-floats nicely 11 | \usepackage{array,tabularx,booktabs} 12 | \usepackage{titletoc} 13 | \usepackage{mathtools} 14 | \usepackage{tikz} 15 | \usepackage[labelfont=bf,textfont=md]{caption} 16 | \usepackage{capt-of} 17 | \usepackage{dsfont} 18 | \usepackage{nicefrac} 19 | \usepackage{amsmath,amssymb,amsthm} 20 | \usepackage{amsfonts} 21 | \usepackage{tcolorbox} 22 | \tcbuselibrary{skins} 23 | \usepackage{cleveref} 24 | \crefformat{figure}{#2{\color{violet}#1}#3} 25 | \usepackage{pgfplots} 26 | \usepgfplotslibrary{colormaps} 27 | \usepackage{fontspec,unicode-math} 28 | \renewcommand{\familydefault}{\sfdefault} 29 | 30 | 31 | %%%%%%%%%%COLORS%%%%%%%%%%%%%%%%%%% 32 | 33 | 34 | \definecolor{dgreen}{RGB}{0,154,84} 35 | \definecolor{sgreen}{RGB}{238,247,244} 36 | \definecolor{zgreen}{RGB}{56,129,93} 37 | 38 | \definecolor{dblue}{RGB}{0,110,185} 39 | \definecolor{sblue}{RGB}{240,243,250} 40 | \definecolor{zblue}{RGB}{65,106,150} 41 | \definecolor{nblue}{RGB}{212,244,245} 42 | \definecolor{tblue}{RGB}{0,163,211} 43 | 44 | \definecolor{nred}{RGB}{245,213,212} 45 | \definecolor{tred}{RGB}{211,48,0} 46 | 47 | \definecolor{royalpurple}{RGB}{120,81,169} 48 | 49 | 50 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 51 | 52 | %%%%%%%%%%%%%NEWCOMMANDS%%%%%%%%%%%%%% 53 | 54 | \newcommand{\dis}{\displaystyle} 55 | \newcommand{\bi}{\begin{itemize}} 56 | \newcommand{\ei}{\end{itemize}} 57 | \newcommand{\bn}{\begin{enumerate}} 58 | \newcommand{\en}{\end{enumerate}} 59 | 60 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 61 | 62 | 63 | 64 | 65 | %%%%%%%%%GEOMETRY%%%%%%%%%%%% 66 | 67 | \RequirePackage{geometry} 68 | \geometry{ 69 | paperwidth=210mm, 70 | paperheight=297mm, 71 | left=42pt, 72 | top=72pt, 73 | textwidth=350pt, 74 | marginparsep=20pt, 75 | marginparwidth=140pt, 76 | textheight=650pt, 77 | footskip=40pt, 78 | } 79 | 80 | 81 | 82 | 83 | 84 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 85 | 86 | 87 | %%%%%%%%%%LETTER SPACING%%%%%%%%%%%%%%%%% 88 | \renewcommand{\normalsize}{\fontsize{10pt}{13pt}\selectfont}% 89 | \renewcommand{\footnotesize}{\fontsize{8pt}{10pt}\selectfont}% 90 | % fullwidth environment, text across textwidth+marginparsep+marginparwidth 91 | \newlength{\overhang} 92 | \setlength{\overhang}{\marginparwidth} 93 | \addtolength{\overhang}{\marginparsep} 94 | % 95 | \newenvironment{fullwidth} 96 | {\ifthenelse{\boolean{@twoside}}% 97 | {\begin{adjustwidth*}{}{-\overhang}}% 98 | {\begin{adjustwidth}{}{-\overhang}}% 99 | }% 100 | {\ifthenelse{\boolean{@twoside}}% 101 | {\end{adjustwidth*}}% 102 | {\end{adjustwidth}}% 103 | } 104 | 105 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 106 | 107 | 108 | 109 | %%%%%%%%%%%CHAPTER HEADER %%%%%%%%%%%%%%%%%% 110 | 111 | \renewcommand{\sffamily}{\color{TFFrameColor}} 112 | \renewcommand\LettrineTextFont{\color{TFFrameColor}\scshape} 113 | \usepackage[x11names]{xcolor} 114 | \newcommand*\ftsize[1]{\fontsize{#1pt}{\numexpr 1.2*#1\relax pt}\selectfont} 115 | \usepackage{colortbl} 116 | \usepackage{framed} 117 | \colorlet{TFFrameColor}{DodgerBlue3} 118 | \renewenvironment{leftbar}{% 119 | \def\FrameCommand{{\color{TFFrameColor}\vrule width 3pt} \hspace{12pt}}% 120 | \MakeFramed {\advance\hsize-\width \FrameRestore}}% 121 | {\endMakeFramed} 122 | \usepackage[explicit,newlinetospace]{titlesec} 123 | 124 | % New command to set the chapter abstract 125 | \newcommand{\chapterabstracttext}{} 126 | 127 | \newcommand{\setchapterabstract}[1]{% 128 | \renewcommand{\chapterabstracttext}{\textit{Chapter abstract: } #1}% 129 | } 130 | 131 | \titleformat{\chapter}[hang]% 132 | {\usefont{T1}{phv}{m}{n}}% Font settings 133 | {% 134 | \parbox[t]{\dimexpr0.12\linewidth-20pt\relax}{\fontsize{48}{48}\selectfont\raisebox{-1.25\height}{\color{TFFrameColor}\thechapter}}% 135 | }% 136 | {1em}% 137 | {% 138 | \begin{minipage}[t]{1.36\textwidth}% Set to full text width 139 | \begin{leftbar}% 140 | {\bfseries\fontsize{24}{30}\selectfont\color{TFFrameColor}% 141 | \rule{0pt}{2ex}\strut #1\hfil\vskip2ex\break% 142 | }% 143 | \chapterabstracttext% 144 | \rule[-1.5ex]{0pt}{1.5ex}% 145 | \end{leftbar}% 146 | \end{minipage}% 147 | } 148 | \titlespacing{\chapter}{0pt}{2\baselineskip}{6\baselineskip} 149 | 150 | 151 | 152 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 153 | 154 | 155 | 156 | %%%%%%%%%%%%%SECTION HEADER%%%%%%%%%%%%%%%%%% 157 | 158 | 159 | 160 | \titleformat{\section} 161 | {\LettrineTextFont\Large\bfseries\color{black}} 162 | {\thesection} 163 | {1em} 164 | {#1} 165 | [{\color{DodgerBlue3}\titlerule[0.8pt]}] 166 | 167 | 168 | \titleformat{\subsection} 169 | {\LettrineTextFont\Large\bfseries\color{black}} 170 | {\thesubsection} 171 | {1em} 172 | {#1} 173 | [{\color{DodgerBlue3}\titlerule[0.8pt]}] 174 | 175 | 176 | 177 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 178 | 179 | 180 | %%%%%%%%%%%%%%SIDE NOTE%%%%%%%%%%%%%%%%%% 181 | 182 | \renewcommand*{\raggedleftmarginnote}{\noindent} 183 | \renewcommand*{\raggedrightmarginnote}{\noindent} 184 | \newcommand{\mn}[1]{\textsuperscript{\thesidenote}{}\marginnote{\textsuperscript{\thesidenote}{}\itshape\footnotesize #1}\refstepcounter{sidenote}} 185 | \newcommand{\fn}[1]{\marginnote{\footnotesize{#1}}} 186 | \newcommand{\lec}[2]{{\setlength{\marginparwidth}{.07\paperwidth}\reversemarginpar\marginnote{\centering\footnotesize{\textsf{\bfseries #1}}\\#2}}} 187 | \newcommand{\sn}[1]{\sidenote{\footnotesize #1}} 188 | 189 | 190 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 191 | 192 | 193 | %%%%%%%%%%%%%%%%%%MARGIN TOC%%%%%%%%%% 194 | 195 | \newcommand{\chaptoc}{ 196 | \startcontents[chapters] 197 | \marginnote{ 198 | \begin{minipage}[b,inner sep=0,outer sep=0]{\linewidth} 199 | { 200 | \bfseries 201 | \sffamily 202 | \centerline{Outline} 203 | } 204 | \vspace{0.5em} 205 | \hrule 206 | \vspace{0.5em} 207 | {\footnotesize % Change the font size here to small 208 | \printcontents[chapters]{}{1}{} 209 | } 210 | \vspace{0.5em} 211 | \hrule 212 | \end{minipage} 213 | }} 214 | 215 | 216 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 217 | 218 | 219 | 220 | %%%%%%%%%%%%%%%%THEOREMBOXES%%%%%%%%%%%%%%%%%%%% 221 | 222 | 223 | 224 | \NewTotalTColorBox[auto counter]{\Example}{+m +m}{ 225 | notitle, 226 | colback=sgreen, 227 | colbacklower=white, 228 | frame hidden, 229 | boxrule=0pt, 230 | bicolor, 231 | sharp corners, 232 | borderline west={4pt}{0pt}{dgreen}, 233 | }{ 234 | \textcolor{black}{ 235 | \sffamily 236 | \textbf{{\color{dgreen}Example~\thetcbcounter.}} \\ 237 | {\color{black} #1} 238 | }% 239 | } 240 | 241 | 242 | 243 | 244 | \NewTotalTColorBox[auto counter]{\Definition}{+m +m +m}{ 245 | notitle, 246 | colback=sblue, 247 | colbacklower=white, 248 | frame hidden, 249 | boxrule=0pt, 250 | bicolor, 251 | sharp corners, 252 | borderline west={4pt}{0pt}{DodgerBlue3}, 253 | }{ 254 | \textcolor{black}{ 255 | \sffamily 256 | \textbf{{\color{DodgerBlue3}Definition~\thetcbcounter:}} % Definition counter 257 | \textbf{{\color{DodgerBlue3}#2}} \\ % Add definition name here 258 | {\color{black} #1} % Definition content 259 | }% 260 | } 261 | 262 | 263 | 264 | \NewTotalTColorBox[auto counter]{\Remark}{+m +m}{ 265 | notitle, 266 | colback=orange!5, 267 | colbacklower=white, 268 | frame hidden, 269 | boxrule=0pt, 270 | bicolor, 271 | sharp corners, 272 | borderline west={4pt}{0pt}{orange}, 273 | }{ 274 | \textcolor{black}{ 275 | \sffamily 276 | \textbf{{\color{orange}Remark~\thetcbcounter.}} \\ 277 | {\color{black} #1} 278 | }% 279 | } 280 | 281 | 282 | 283 | 284 | 285 | \NewTotalTColorBox[auto counter]{\Proposition}{+m +m}{ 286 | notitle, 287 | colback=royalpurple!7, 288 | colbacklower=white, 289 | frame hidden, 290 | boxrule=0pt, 291 | bicolor, 292 | sharp corners, 293 | borderline west={4pt}{0pt}{royalpurple}, 294 | }{ 295 | \textcolor{royalpurple}{ 296 | \sffamily 297 | \textbf{{\color{royalpurple}Proposition~\thetcbcounter.}} \\% 298 | }% 299 | {\color{black}#1} 300 | \tcblower% 301 | \textcolor{royalpurple}{ 302 | \sffamily 303 | \textbf{{\color{royalpurple}Proof.}$\;$}% 304 | }% 305 | #2 306 | } 307 | -------------------------------------------------------------------------------- /lecture2.tex: -------------------------------------------------------------------------------- 1 | \setchapterabstract{This chapter is a foundational concept in probability theory that allows us to assess the likelihood of one event occurring given that another event has already happened. This chapter delves into the key principles and applications of conditional probability. We begin by exploring the fundamental difference between ordinary probability and conditional probability, emphasizing the importance of contextual dependence.} 2 | \chapter{Conditional Probability} 3 | \vspace{-1.5cm} 4 | 5 | 6 | 7 | 8 | %%%%%%INSERT TOC BELOW 1ST SECTION%%%%%%%%%%%% 9 | 10 | {\chaptoc\noindent\begin{minipage}[inner sep=0,outer sep=0]{0.9\linewidth}\section{Conditional Probability}\end{minipage}} 11 | 12 | 13 | In our previous lecture, we explored the fundamental concepts of probability theory. We established the core axioms that govern probabilities and their applications in calculating the likelihood of events. However, the world around us is rarely as simple as independent events happening in isolation. Often, the likelihood of one event occurring is influenced by the knowledge of another event having already happened 14 | 15 | \Example{Consider a sample space that includes three disjoint events, $A$, $B$, and $C$, as illustrated in Figure \cref{fig:condprobsimp}. Suppose we receive new information confirming that event $A$ has indeed occurred, but we do not have any updates about events $B$ and $C$. How does this additional information about event $A$ affect our original probability model? 16 | } 17 | 18 | Since events $A$, $B$, and $C$ are disjoint (as shown\sn{\input{condprobsimp}\captionof{figure}{Visual representation of the conditional probability of event $B$ and $C$ given event A. When $A$ occurs, the probability of $B$ and $C$ also occurring is zero, since the three events are mutually exclusive.}\label{fig:condprobsimp}} in Figure \cref{fig:condprobsimp}), they cannot happen at the same time. This means that if event $A$ occurs, events $B$ and $C$ cannot occur. Events $B$ and $C$ together form the complement of event $A$, denoted as $B \cup C = A^{c}$. Therefore, if $A$ is observed, $A^c$ cannot be observed, i.e. 19 | \[ 20 | \mathds{P}(A^{c})=\mathds{P}(B \cup C)\overset{\text{(A3)}}{=}\mathds{P}(B) + \mathds{P}(C) = 0 21 | \] 22 | If and only if $\mathds{P}(B)=\mathds{P}(C)=0$. 23 | 24 | In this modified sample space, the probability distribution is also updated. The probability of event $A$ occurring is now $1$, which makes it a new sample space by axiom (A2), i.e. $\widehat{\Omega}\triangleq A$ , since it is the only event in the new sample space. 25 | 26 | \Example{ 27 | Consider a sample space that includes three events, $A$, $B$, and $C$, of which event $C$ is disjoint from events $A$ and $B$, but events $A$ and $B$ are not disjoint as illustrated in Figure \cref{fig:condprobvis}. Suppose we receive new information confirming that event $A$ has indeed occurred, but we do not have any updates about events $B$ and $C$. How does this additional information about event $A$ affect our original probability model? 28 | } 29 | 30 | When we confirm that event $A$ has occurred, we must revise our probability model to reflect this new information as done in the previous example. Since event $C$ is mutually exclusive with events $A$ and $B$, we can immediately conclude that event $C$ can not occur, i.e. $\mathds{P}(C)=0$. 31 | 32 | Event\sn{\input{condprobvis}\captionof{figure}{Visual representation of the conditional probability of event $B$ given event $A$. When A occurs, the probability of $B$ also occurring is equivalent to the intersection between $A$ and $B$.}\label{fig:condprobvis}}$B$ can be broken down into two parts: the region where $B$ intersects with $A$ (the green region in Figure \cref{fig:condprobvis}), and the region where $B$ is observed without $A$. 33 | \[ 34 | B=(A\cap B)\cup (A^{c}\cap B) 35 | \] 36 | However, since event $A$ is assumed to be observed, the latter region is no longer possible, i.e. $\mathds{P}(A^{c}\cap B)=0$. Therefore, the probability of event $B$ is reduced to 37 | \[ 38 | \mathds{P}(B) \overset{\text{(A3)}}{=} \mathds{P}(A \cap B) + \mathds{P}(A^c \cap B) = \mathds{P}(A \cap B) 39 | \] 40 | Now, we need to establish the relationship between the probability of event $A$ and the probability of the intersection of events $A$ and $B$, denoted as $A \cap B$. In the revised model, event $A$ becomes the new sample space, and accordingly we denote it by $\widehat{\Omega}$. Moreover, we denote the intersection of events $A$ and $B$ as $\widehat{B}$ in the revised model. The problem is now equivalent to calculating the probability of $\widehat{B}$ within the new sample space $\widehat{\Omega}$. This is nothing more than a classic demonstration of the discrete uniform law: 41 | \[ 42 | \begin{array}{ccl} 43 | \dis\frac{|\widehat{B}|}{|\widehat{\Omega}|} & = & \dis\frac{|A \cap B|}{|A|} \\[0.4cm] 44 | & = & \dis\frac{\dis\nicefrac{\dis|A \cap B|}{\dis|\Omega|}}{\dis\nicefrac{\dis|A|}{\dis|\Omega|}} \\[0.4cm] 45 | & = & \dis\frac{\mathds{P}(A \cap B)}{\mathds{P}(A)} 46 | \end{array} 47 | \] 48 | This ratio represents the probability of event $B$ occurring given that event $A$ has occurred. We define this as the conditional probability of $B$ conditioned on $A$. 49 | 50 | \Definition{ 51 | Let $A,B\subseteq\Omega$, the conditional probability of an event $B$ conditioned on an event $A$ is given by 52 | \[ 53 | \mathds{P}(B\mid A)\triangleq\frac{\mathds{P}(A \cap B)}{\mathds{P}(A)} 54 | \] 55 | }{Conditional Probability} 56 | 57 | 58 | Referring to Figure \ref{fig:condprobvis}, we can observe that $|A\cap B|= 3$ and $|A| = 8$. Using these values, we can calculate the conditional probability of $B$ given $A$ as: 59 | \[ 60 | \begin{array}{ccl} 61 | \mathds{P}(B \mid A) &=& \dis\frac{P(A \cap B)}{P(A)} \\[0.4cm] 62 | &=& \dis\frac{|A \cap B|}{|A|}\\[0.4cm] 63 | &=& \dis\frac{3}{8} 64 | \end{array} 65 | \] 66 | 67 | \clearpage 68 | \section{Die Roll Example} 69 | We consider a simple example where we utilize the conditional probability. 70 | \Example{ 71 | Consider two four-faced dice. Let $B$ denote the event that one of the two die resulted in a value of $2$, and the other die resulted in a value of at least $2$. Moreover, define $M\triangleq\max(X,Y)$. What is the probability that $M=3$, where one of the dices has an attained value of $2$? 72 | } 73 | 74 | In this example, there are 16 possible outcomes, all equally likely. We define event $B$ as 75 | \[ 76 | B \triangleq \{(x, y)\in X\times Y \mid \min(x, y) = 2\} 77 | \] 78 | 79 | meaning one die shows a value of 2 and the other shows a value of least a 2. Figure \cref{fig:condprobexdice} illustrates\sn{\input{condprobexdice}\captionof{figure}{The sample space in this problem is reduced upon the occurrence of the event $B$ (highlighted in red).}\label{fig:condprobexdice}} these scenarios, we can see that event $B$ consists of the outcomes 80 | \[ 81 | B=\{(2,2), (2,3), (2,4), (3,2), (4,2)\} 82 | \] 83 | Now, we want to find the probability that $M=3$ (i.e., the maximum value is 3) given that event $B$ has occurred. Out of the 5 scenarios for event $B$, only 2 of them result in an observation for $M$: 84 | \[ 85 | M=\{(2,3), (3,2)\} 86 | \] 87 | Therefore, the conditional probability is calculated as 88 | \[ 89 | \mathds{P}(M=3 \mid B) = \frac{2}{5}. 90 | \] 91 | 92 | Alternatively, we can directly apply the conditional probability formula to reach this result. In this case, we have: 93 | \[ 94 | \mathds{P}(M=3\mid B)=\frac{\mathds{P}((M=3)\cap B)}{\mathds{P}(B)}=\frac{\nicefrac{2}{16}}{\nicefrac{5}{16}}=\frac{2}{5} 95 | \] 96 | 97 | \Remark{Conditional probabilities are same as ordinary probabilities, but applied to different situations. Intrinsically speaking, they obey the standard probability axioms. 98 | } 99 | 100 | The only crucial difference lies in the consideration of additional information. Ordinary probability treats all events independently, while conditional probability takes into account the knowledge that another event has already transpired. This context-dependence allows conditional probability to capture more nuanced relationships between events. 101 | 102 | 103 | \section{Properties of Conditional Probability} 104 | Conditional probability is a fundamental concept in probability theory that allows us to update our beliefs about the likelihood of events based on new information. In this section, we will expand our knowledge on conditional probabilities by developing some basic properties. 105 | 106 | We begin by emphasizing that conditional probability adheres to the same basic principles as ordinary probability, namely the probability axioms. 107 | \Proposition{Conditional probability satisfies the probability axioms. 108 | } 109 | {We shall demonstrate this for each axiom: 110 | \bi 111 | \item[(A1)] $\mathds{P}(A\mid B)\geqslant 0$, since 112 | \[ 113 | \mathds{P}(A\mid B)=\frac{\mathds{P}(A\cap B)\geqslant 0}{\mathds{P}(B)>0}\geqslant 0 114 | \] 115 | \item[(A2)] $\mathds{P}(\Omega\mid B)=1$, since 116 | \[ 117 | \mathds{P}(\Omega\mid B)=\frac{\mathds{P}(\Omega\cap B)}{\mathds{P}(B)}=\frac{\mathds{P}(B)}{\mathds{P}(B)}=1 118 | \] 119 | \item[(A3)] Assume $A\cap C=\emptyset$, then 120 | \[ 121 | \begin{array}{ccl} 122 | \mathds{P}(A\cup C\mid B)&=&\dis\frac{\mathds{P}\big((A\cup C)\cap B\big)}{\mathds{P}(B)} \\[0.4cm] 123 | &=&\dis\frac{\mathds{P}\big((A\cap B)\cup(C\cap B)\big)}{\mathds{P}(B)} \\[0.4cm] 124 | &=&\dis\frac{\mathds{P}(A\cap B)}{\mathds{P}(B)} + \frac{\mathds{P}(C\cap B)}{\mathds{P}(B)} \\[0.4cm] 125 | &=&\dis\mathds{P}(A\mid B)+\mathds{P}(C\mid B) 126 | \end{array} 127 | \] 128 | Note that the pair of $A$ that belongs in $B$ is also disjoint from the part of $C$ that belongs to $B$. 129 | \ei 130 | } 131 | Thus, this ensures that our calculations and interpretations remain consistent within the framework of probability theory. 132 | 133 | 134 | 135 | -------------------------------------------------------------------------------- /condprobsimp.tex: -------------------------------------------------------------------------------- 1 | \begin{center} 2 | 3 | 4 | \tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt 5 | 6 | \begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] 7 | %uncomment if require: \path (0,452); %set diagram left start at 0, and has height of 452 8 | 9 | %Shape: Rectangle [id:dp3366248195371335] 10 | \draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][line width=1.5] (8.67,44.29) -- (203.94,44.29) -- (203.94,171.39) -- (8.67,171.39) -- cycle ; 11 | %Shape: Ellipse [id:dp6862156891277029] 12 | \draw [color={rgb, 255:red, 0; green, 163; blue, 211 } ,draw opacity=1 ][fill={rgb, 255:red, 212; green, 244; blue, 245 } ,fill opacity=1 ][line width=1.5] (32.57,89.66) .. controls (43.83,65.33) and (66.18,52.63) .. 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(177.36,31.77) ; 59 | \draw [shift={(174.05,29.95)}, rotate = 29.13] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (9.91,-4.76) -- (0,0) -- (9.91,4.76) -- (6.58,0) -- cycle ; 60 | %Straight Lines [id:da04618455972416813] 61 | \draw [line width=1.5] (105,179) -- (105,216) ; 62 | \draw [shift={(105,220)}, rotate = 270] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (11.07,-5.32) -- (0,0) -- (11.07,5.32) -- (7.35,0) -- cycle ; 63 | %Shape: Rectangle [id:dp6211196777653878] 64 | \draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][line width=1.5] (9.11,225.69) -- (204.38,225.69) -- (204.38,352.79) -- (9.11,352.79) -- cycle ; 65 | %Shape: Ellipse [id:dp8000435370413392] 66 | \draw [color={rgb, 255:red, 0; green, 163; blue, 211 } ,draw opacity=1 ][fill={rgb, 255:red, 212; green, 244; blue, 245 } ,fill opacity=1 ][line width=1.5] (33.01,271.06) .. controls (44.27,246.73) and (66.62,234.03) .. (82.94,242.69) .. controls (99.26,251.35) and (103.36,278.09) .. (92.11,302.42) .. controls (80.85,326.74) and (58.5,339.44) .. (42.18,330.79) .. controls (25.86,322.13) and (21.76,295.39) .. (33.01,271.06) -- cycle ; 67 | %Shape: Ellipse [id:dp03731012601422612] 68 | \draw [color={rgb, 255:red, 211; green, 48; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 211; green, 48; blue, 0 } ,fill opacity=1 ] (51.33,259.42) .. controls (51.33,258.44) and (52.07,257.65) .. (52.98,257.65) .. controls (53.9,257.65) and (54.64,258.44) .. (54.64,259.42) .. controls (54.64,260.39) and (53.9,261.19) .. (52.98,261.19) .. controls (52.07,261.19) and (51.33,260.39) .. (51.33,259.42) -- cycle ; 69 | %Shape: Ellipse [id:dp7662796034721766] 70 | \draw [color={rgb, 255:red, 211; green, 48; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 211; green, 48; blue, 0 } ,fill opacity=1 ] (63.91,272.89) .. controls (63.91,271.91) and (64.65,271.12) .. (65.56,271.12) .. controls (66.48,271.12) and (67.22,271.91) .. (67.22,272.89) .. controls (67.22,273.87) and (66.48,274.66) .. (65.56,274.66) .. controls (64.65,274.66) and (63.91,273.87) .. (63.91,272.89) -- cycle ; 71 | %Shape: Ellipse [id:dp3428571943128822] 72 | \draw [color={rgb, 255:red, 211; green, 48; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 211; green, 48; blue, 0 } ,fill opacity=1 ] (63.91,272.89) .. controls (63.91,271.91) and (64.65,271.12) .. (65.56,271.12) .. controls (66.48,271.12) and (67.22,271.91) .. (67.22,272.89) .. controls (67.22,273.87) and (66.48,274.66) .. (65.56,274.66) .. controls (64.65,274.66) and (63.91,273.87) .. (63.91,272.89) -- cycle ; 73 | %Shape: Ellipse [id:dp9003120016223698] 74 | \draw [color={rgb, 255:red, 211; green, 48; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 211; green, 48; blue, 0 } ,fill opacity=1 ] (48.18,286.36) .. controls (48.18,285.38) and (48.92,284.59) .. (49.84,284.59) .. controls (50.75,284.59) and (51.49,285.38) .. (51.49,286.36) .. controls (51.49,287.34) and (50.75,288.13) .. (49.84,288.13) .. controls (48.92,288.13) and (48.18,287.34) .. (48.18,286.36) -- cycle ; 75 | %Shape: Ellipse [id:dp3839640412271208] 76 | \draw [color={rgb, 255:red, 211; green, 48; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 211; green, 48; blue, 0 } ,fill opacity=1 ] (74.6,299.83) .. controls (74.6,298.86) and (75.34,298.06) .. (76.26,298.06) .. controls (77.17,298.06) and (77.91,298.86) .. (77.91,299.83) .. controls (77.91,300.81) and (77.17,301.6) .. (76.26,301.6) .. controls (75.34,301.6) and (74.6,300.81) .. (74.6,299.83) -- cycle ; 77 | %Shape: Ellipse [id:dp6698541861451095] 78 | \draw [color={rgb, 255:red, 211; green, 48; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 211; green, 48; blue, 0 } ,fill opacity=1 ] (54.48,311.28) .. controls (54.48,310.31) and (55.22,309.51) .. (56.13,309.51) .. controls (57.04,309.51) and (57.78,310.31) .. (57.78,311.28) .. controls (57.78,312.26) and (57.04,313.05) .. (56.13,313.05) .. controls (55.22,313.05) and (54.48,312.26) .. 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(75.23,254.03) -- cycle ; 83 | 84 | % Text Node 85 | \draw (8.88,22.37) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$\Omega $}; 86 | % Text Node 87 | \draw (13.88,52.5) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$A$}; 88 | % Text Node 89 | \draw (161.81,56.09) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$B$}; 90 | % Text Node 91 | \draw (171.13,89.67) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$C$}; 92 | % Text Node 93 | \draw (76.89,14.53) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$\mathds{P}\left(A_{1}^{c}\right)=0$}; 94 | % Text Node 95 | \draw (14.32,233.9) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$A$}; 96 | % Text Node 97 | \draw (12.93,197.99) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$\widehat{\Omega}$}; 98 | 99 | 100 | \end{tikzpicture} 101 | \end{center} 102 | -------------------------------------------------------------------------------- /condprobvis.tex: -------------------------------------------------------------------------------- 1 | \begin{center} 2 | 3 | 4 | \tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt 5 | 6 | \begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] 7 | %uncomment if require: \path (0,397); %set diagram left start at 0, and has height of 397 8 | 9 | %Straight Lines [id:da04618455972416813] 10 | \draw [line width=1.5] (105,179) -- (105,216) ; 11 | \draw [shift={(105,220)}, rotate = 270] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (11.07,-5.32) -- (0,0) -- (11.07,5.32) -- (7.35,0) -- cycle ; 12 | %Shape: Rectangle [id:dp6211196777653878] 13 | \draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][line width=1.5] (9.11,225.69) -- (204.38,225.69) -- (204.38,352.79) -- (9.11,352.79) -- cycle ; 14 | %Shape: Rectangle [id:dp9414156037094199] 15 | \draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][line width=1.5] (8.79,44.29) -- (204.06,44.29) -- (204.06,171.39) -- (8.79,171.39) -- cycle ; 16 | %Shape: Ellipse [id:dp00230890425695085] 17 | \draw [color={rgb, 255:red, 0; green, 163; blue, 211 } ,draw opacity=1 ][fill={rgb, 255:red, 212; green, 244; blue, 245 } ,fill opacity=1 ][line width=1.5] (32.69,89.66) .. controls (43.95,65.33) and (66.3,52.63) .. 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(174.6,97.66) ; 90 | \draw [shift={(175.24,101.44)}, rotate = 260.85] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (9.91,-4.76) -- (0,0) -- (9.91,4.76) -- (6.58,0) -- cycle ; 91 | %Curve Lines [id:da9554191414975919] 92 | \draw [line width=1.5] (99.42,135.57) .. controls (111.22,131.55) and (130.52,133.6) .. (144.3,143.43) ; 93 | \draw [shift={(147.31,145.78)}, rotate = 220.55] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (9.91,-4.76) -- (0,0) -- (9.91,4.76) -- (6.58,0) -- cycle ; 94 | %Curve Lines [id:da6195367761893151] 95 | \draw [line width=1.5] (77.42,315.57) .. controls (89.22,311.55) and (108.52,313.6) .. (122.3,323.43) ; 96 | \draw [shift={(125.31,325.78)}, rotate = 220.55] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (9.91,-4.76) -- (0,0) -- (9.91,4.76) -- (6.58,0) -- cycle ; 97 | 98 | % Text Node 99 | \draw (12.93,197.99) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$\widehat{\Omega }$}; 100 | % Text Node 101 | \draw (9,22.37) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$\Omega $}; 102 | % Text Node 103 | \draw (20.89,53.56) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$A$}; 104 | % Text Node 105 | \draw (22.58,242.9) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$A$}; 106 | % Text Node 107 | \draw (95.19,323.13) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$\widehat{B} \triangleq A\cap B$}; 108 | % Text Node 109 | \draw (122,100.59) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {$\mathds{P}(C) =0$}; 110 | % Text Node 111 | \draw (109.52,147.78) node [anchor=north west][inner sep=0.75pt] [xscale=1.5,yscale=1.5] {{\tiny$\mathds{P}(B\cap A^{c})=0$}}; 112 | 113 | 114 | \end{tikzpicture} 115 | \end{center} 116 | --------------------------------------------------------------------------------