\documentclass[a4paper,12pt]{article}
\begin{document}
\parindent=0pt

QUESTION

 An electronic system receives signals as input and sends out
   appropriate coded messages as output.

   The system consists of 3 converters $C_1,C_2$ and $C_3$, 2
   monitors $M_1$ and $M_2$ and a perfectly reliable three way
   switch for connecting the input to the converters. The incoming
   signal is changed into a code by one of the converters and the
   monitors check whether the conversion is correct

   Initially the signal is fed into $C_1$. If $M_1$ passes the
   conversion the coded message is sent out. If $M_!$ rejects the
   conversion the input is switched to $C_2$ and the conversion is
   checked by $M_2$. If $M_2$ passes the conversion, the coded
   message is sent out. If $M_2$ rejects the conversion, the input
   is switched to $C_3$ and the coded message is sent out without
   any further checks.

   Each of the converters has probability 0.9 of correctly coding
   the incoming message. Each of the monitors has probability 0.8
   of rejecting a wrongly coded message and also probability 0.8
   of passing a correctly coded message.

   Draw a probability tree and hence show that the probability of
   a correct output from the system is about 0.968.
\bigskip

ANSWER

 \setlength{\unitlength}{.2in}

\begin{picture}(22,19)

\put(0,8.5){$\bullet$} \put(0.1,8.6){\line(1,2){2}}
\put(2.2,12.5){$C_{1\surd}$} \put(0.6,11){$.9$}
\put(0.1,8.6){\line(1,-2){2}} \put(2.2,4.5){$C_{1\times}$}
\put(0.6,6){$.1$}

\put(3.5,13){\line(1,1){2}} \put(5.7,15){$M_{1\surd}$}
\put(6.4,15.2){\circle{2}} \put(7.5,15){$A$} \put(4,14.2){$.8$}

\put(3.5,13){\line(1,0){2}} \put(5.7,13){$M_{1\times}$}
\put(4,12.2){$.2$}

\put(3.5,4.8){\line(1,0){2}} \put(5.7,4.5){$M_{1\surd}$}
\put(4,5){$.8$}

\put(3.5,4.8){\line(1,-1){2}}
\put(5.7,2.2){\framebox(1.5,1){$M_{1\times}$}}
\put(7.5,2.4){$\overline{A}$} \put(4,3){$.2$}

\put(7.5,13){\line(4,3){4}} \put(11.7,16){$C_{2\surd}$}
\put(9.3,15){$.8$}

\put(7.5,13){\line(4,-3){4}} \put(11.7,9.8){$C_{2\times}$}
\put(9.3,10.5){$.2$}


\put(7.5,5){\line(4,3){4}} \put(11.7,8){$C_{2\surd}$}
\put(9.3,7){$.8$}

\put(7.5,5){\line(4,-3){4}} \put(11.7,1.8){$C_{2\times}$}
\put(9.3,2.5){$.2$}

\put(13.2,16.8){\line(3,2){2}} \put(15.6,18){$M_{2\surd}$}
\put(16.3,18.2){\circle{2}} \put(17.4,18){$B$} \put(14,18.1){$.8$}

\put(13.2,16.8){\line(3,-2){2.5}} \put(15.6,14.8){$M_{2\times}$}
\put(14,15){$.2$}

\put(13.2,10.3){\line(3,2){2.5}} \put(15.6,12){$M_{2\surd}$}
\put(14,11.2){$.8$}

\put(13.2,10.3){\line(1,0){2}}
\put(15.6,9.8){\framebox(1.5,1){$M_{2\times}$}} \put(14,9.7){$.2$}
\put(17.4,10){$\overline{B}$}

\put(13.2,8.2){\line(1,0){2}} \put(15.6,8){$M_{2\surd}$}
\put(16.3,8.2){\circle{2}} \put(17.4,8){$$} \put(14,8.3){$.8$}
\put(17.4,8){$C$}

\put(13.2,8.2){\line(3,-2){2.5}} \put(15.6,6){$M_{2\times}$}
\put(14,6.7){$.2$}

\put(13.2,2){\line(3,2){2.5}} \put(15.6,3.8){$M_{2\surd}$}
\put(14,3){$.8$}

\put(13.2,2){\line(3,-2){2}}
\put(15.6,0){\framebox(1.5,1){$M_{2\times}$}} \put(14,0.2){$.2$}
\put(17.4,0.2){$\overline{C}$}

\put(17.2,15){\line(3,2){3}} \put(20.5,17){$C_{3\surd}$}
\put(21,17.3){\circle{2}} \put(18.8,16.5){$.8$} \put(22.2,17){$D$}

\put(17.2,15){\line(1,0){3}}
\put(20.3,14.5){\framebox(1.5,1){$C_{3\times}$}}
\put(18.8,14.4){$.2$} \put(22.2,14.7){$\overline{D}$}

\put(17.2,12.5){\line(1,0){3}} \put(20.5,12.2){$C_{3\surd}$}
\put(21,12.5){\circle{2}} \put(18.8,12.9){$.8$}
\put(22.2,12.2){$E$}

\put(17.2,12.5){\line(3,-2){3}}
\put(20.3,10){\framebox(1.5,1){$C_{3\times}$}}
\put(18.8,10.5){$.2$} \put(22.2,10.2){$\overline{E}$}

\put(17.2,6){\line(3,2){3}} \put(20.5,8){$C_{3\surd}$}
\put(21,8.3){\circle{2}} \put(18.8,7.5){$.8$} \put(22.2,8){$F$}

\put(17.2,6){\line(1,0){3}}
\put(20.3,5.5){\framebox(1.5,1){$C_{3\times}$}}
\put(18.8,5.4){$.2$} \put(22.2,5.7){$\overline{F}$}

\put(17.2,3.5){\line(1,0){3}} \put(20.5,3.2){$C_{3\surd}$}
\put(21,3.5){\circle{2}} \put(18.8,3.9){$.8$} \put(22.2,3.2){$G$}

\put(17.2,3.5){\line(3,-2){3}}
\put(20.3,1){\framebox(1.5,1){$C_{3\times}$}} \put(18.8,1.5){$.2$}
\put(22.2,1.2){$\overline{G}$}


\end{picture}

\begin{picture}(22,4)
\put(0,0){$C_{1\surd}\hspace{.3in} C_1$\ converter correct}
\put(0,1){$C_{1\times}\hspace{.3in} C_1$\ converter incorrect}
\put(0,2){$M_{1\surd}\hspace{.3in} M_1$\ monitor correct}
\put(0,3){$M_{1\times}\hspace{.3in} M_1$\ monitor incorrect}

\put(15,0){\framebox(1.5,1){ }\ incorrect outcome}
\put(15.5,2.5){\circle{2}\hspace{.2in} correct outcome}

\end{picture}

\vspace{.5in}
    \begin{tabular}{llll}
      \ &P correct&\ &P incorrect\\
     A &$0.9\times 0.8=0.72$& \={A} & $0.1\times 0.2=0.02$\\
     B&$0.9^2\times 0.2\times 0.8=0.1296$&\={B}&$0.9\times 0.2^2\times 0.1=0.0036$\\
     C&$0.9\times 0.1\times 0.8^2=0.0576$&\={C}&$0.1^2\times 0.8\times 0.2=0.0016$\\
     D&$0.93\times 0.2^2=0.02916$&\={D}&$0.92\times 0.1\times 0.2^2=0.00324$\\
     E&$0.9^2\times 0.2\times 0.1\times 0.8=0.01296$&\={E}&$0.9\times 0.2\times 0.12\times 0.8=0.00144$\\
     F&$0.1\times 0.8\times 0.9^2\times 0.2= 0.1296$&\={F}&$0.12\times 0.8\times 0.9\times 0.2=0.00144$\\
     G&$0.1^2\times 0.8^2\times 0.9=0.00576$&\={G}&$0.1^3\times
     0.8^2=0.00064$\\
     &Total $=0.96804 \approx 0.968$&&Total as check $=0.03196$
    \end{tabular}

\end{document}
