Examples on Value of Information

donanim

$1.$ A new computer virus, called SMASH, produces symptoms very similar to a major hardware failure. You have observed these symptoms in your personal computer, and feel that the cause is equally likely to be SMASH and a genuine failure. There are two different actions you may take. You may either attempt to clean SMASH by purchasing a special$-$purpose virus cleaning program that costs $\backslash (\backslash$$ $200\backslash ),$ or you may do nothing. As a result of these actions, your computer may a$)$ become perfectly operational, b$)$ become operational, but lose some of the programs you had on your hard disk $($you estimate those programs to be worth $\backslash$$$1000),$ or c$)$ crash completely, resulting in a loss of hardware and software that amounts to $\backslash (\backslash$$ $3000\backslash ).$

One problem with the cleaning program is that it aggravates the potential hardware failure and increases the chances of a complete breakdown. You have come up with the following probability estimates:

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|\}$

$\backslash$hline OUTCOME & NO LOSS & SOFTWARELOSS & HARDWARE $\backslash$& $\backslash \backslash$

sOFTWARE LOSs $\backslash \backslash$

$\backslash$hline IF YOU HAVE SMASH AND YOU DO NOTHING & $0\mathrm{.}00$ & $0\mathrm{.}80$ & $0\mathrm{.}20\backslash \backslash$

$\backslash$hline IF YOU HAVE A FAILURE AND YOU DO NOTHING & $0\mathrm{.}00$ & $0\mathrm{.}50$ & $0\mathrm{.}50\backslash \backslash$

$\backslash$hline IF YOU HAVE SMASH AND ATTEMPT TO CLEAN IT & $0\mathrm{.}60$ & $0\mathrm{.}40$ & $0\mathrm{.}00\backslash \backslash$

$\backslash$hline IF YOU HAVEA FAILURE AND ATTEMPT TO CLEAN SMASH & $0\mathrm{.}00$ & $0\mathrm{.}30$ & $0\mathrm{.}70\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

Because SMASH is so new, a single virus$-$scan program tries detecting it$.$ This program costs $\backslash$$$50($it can be used only once$),$ but it is not entirely reliable. If your computer is infected with SMASH, the program will discover this only $\backslash (50\backslash \%\backslash )$ of the time. If$,$ however, your computer is clean, the scanning program will never erroneously diagnose it as infected.

Draw the decision tree for this problem and use decision analysis $($maximizing expected monetary value$)$ to decide whether you should buy either program.