% Mammography problem from 'Intro to Bayes'
% Author: John Henderson
\documentclass[10pt]{article}
\usepackage{tikz}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{5pt}%
\usetikzlibrary{positioning,decorations.pathreplacing,shapes}
\usepackage[english]{babel}
\usepackage{microtype}
\usepackage[hmargin=1.5cm,vmargin=1cm]{geometry}
\usepackage{amsmath}
\DeclareMathOperator{\p}{p}
\newcommand*{\cancer}{\text{cancer}}
\newcommand*{\testp}{\text{test}+}
\begin{document}
\begin{tikzpicture}[%
  % common options for blocks:
  block/.style = {draw, fill=blue!30, align=center, anchor=west,
              minimum height=0.65cm, inner sep=0},
  % common options for the circles:
  ball/.style = {circle, draw, align=center, anchor=north, inner sep=0}]

% circle illustrating all women
\node[ball,text width=3cm,fill=purple!20] (all) at (6,0) {All women};

% two circles showing split of p{cancer} and p{~cancer}
\node[ball,fill=red!70,text width=0.1cm,anchor=base] (pcan) at (3.5,-5.5) {};
\node[ball,fill=blue!40,text width=2.9cm,anchor=base] (pncan) at (8.5,-6)
   {Women without cancer\\
    $\p({\sim}\cancer) = 99\%$};

% arrows showing split from all women to cancer and ~cancer
\draw[->,thick,draw=red!50] (all.south) to [out=270,in=90] (pcan.north);
\draw[->,thick,draw=blue!80] (all.south) to [out=270,in=110] (pncan.100);

% transition from all women to actual cancer rates
\node[anchor=north,text width=10cm,inner sep=.05cm,align=center,fill=white]
  (why1) at (6,-3.7) {In measuring, we find:};

% note illustration the p{cancer} circle (text won't fit inside)
\node[inner sep=0,anchor=east,text width=3.3cm] (note1) at (3.2,-5.5) {
   Women with cancer $\p(\cancer) = 1\%$};

% draw the sieves
\node[block,anchor=north,text width=4.4cm,fill=green!50] (tray1) at
   (3.5,-8.8) {\small{$\p(\testp\mid\cancer)=0.8$}};

\node[block,anchor=north,text width=4.4cm,fill=green!50] (tray2) at
   (8.5,-8.8) {$\p(\testp\mid{\sim}\cancer)=0.096$};

% text explaining how p{cancer} and p{~cancer} behave as they
% pass through the sieves
\node[anchor=west,text width=6cm] (note1) at (-6,-9.1) {
   Now we pass both groups through the sieve; note that both
     sieves are \emph{the same}; they just behave differently
     depending on which group is passing through. \\ 
     Let $\testp=$ a positve mammography.};

% arrows showing the circles passing through the seives
\draw[->,thick,draw=red!80] (3.5,-5.9) -- (3.5,-8.6);
\draw[->,thick,draw=blue!50] (8.5,-8.1) -- (8.5,-8.6);

% numerator
\node[ball,text width=0.05cm,fill=red!70] (can) at (6,-10.5) {};

% dividing line
\draw[thick] (5,-11) -- (7,-11);

% demoniator
\node[ball,text width=0.39cm,fill=blue!40,anchor=base] (ncan) at (6.5,-11.5) {};
\node[ball,text width=0.05cm,fill=red!70,anchor=base] (can2) at (5.5,-11.5) {};

% plus sign in denominator
\draw[thick] (5.9,-11.4) -- (5.9,-11.6);
\draw[thick] (5.8,-11.5) -- (6,-11.5);

% arrows showing the output of the sieves formed the fraction
\draw[->,thick,draw=red!80] (tray1.south) to [out=280,in=180] (can);
\draw[->,thick,draw=red!80] (tray1.south) to [out=280,in=180] (can2);
\node[anchor=north,inner sep=.1cm,align=center,fill=white] (why2) at
   (3.8,-9.8) {$1\% * 80\%$};

\draw[->,thick,draw=blue!50] (tray2.south) to [out=265,in=0] (ncan);
\node[anchor=north,inner sep=.1cm,align=center,fill=white] (why2) at
   (8.4,-9.8) {$99\% * 9.6\%$};

% explanation of final formula
\node[anchor=north west,text width=6.5cm] (note2) at (-6,-12.5)
   {Finally, to find the probability that a positive test
       \emph{actually means cancer}, we look at those who passed
       through the sieve \emph{with cancer}, and divide by all who
       received a positive test, cancer or not.}; 

% illustrated fraction turned into math
\node[anchor=north,text width=10cm] (solution) at (6,-12.5) {
  \begin{align*}
      \frac{\p(\testp\mid\cancer)}{\p(\testp\mid\cancer)
        + \p(\testp\mid{\sim}\cancer)} &= \\
      \frac{1\% * 80\%}{(1\% * 80\%) + (99\% * 9.6\%)} &= 7.8\%
        = \p(\cancer\mid\testp)
   \end{align*}};
\end{tikzpicture}
\end{document}