Cost$-$to$-$loss ratio problem. Consider the decision problem shown in Figure $5\mathrm{.}26.$ This basic decision tree often is called a cost$-$to$-$loss ratio problem and is characterized as a decision situation in which the question is whether to take some protective action in the face of possible adverse circumstances. For example, the umbrella problem $($Figure $4\mathrm{.}9)$ is a cost$-$to$-$loss ratio problem. Taking the umbrella incurs a fixed cost and protects against possible adverse weather. A farmer may face a cost$-$to$-$loss ratio problem if there is a threat of freezing weather that could damage a fruit crop. Steps can be taken to protect the orchard, but they are costly$.$ If no steps are taken, the air temperature may or may not become cold enough to damage the crop.

Sensitivity analysis is easily performed for the cost$-$to$-$loss ratio problem. How large can the probability $\backslash ($ p $\backslash )$ become before "Take Protective Action" becomes the optimal $($minimum expected cost$)$ alternative? Given your answer, what kind of information does the decision maker need in order to make the decision? $($Hint: This is an algebra problem. If that makes you uncomfortable, substitute numerical values for $\backslash ($ C$,$ L $\backslash ),$ and $\backslash ($ p $\backslash ).$