The managers of Red Valley Auto Products are considering the national launch of a new car$-$

cleaning product. For simplicity, the potential average sales of the product during its lifetime

are classified as being either high, medium or low and the net present value of the product under

each of these conditions is estimated to be $$80$ million, $$15$ million and $-40$ million,

respectively. The company's marketing manager estimates that there is a $0\mathrm{.}3$ probability that

average sales will be high, a $0\mathrm{.}4$ probability that they will be medium and a $0\mathrm{.}3$ probability that

they will be low. It can be assumed that the company's objective is to maximize expected net

present value.

a$)$ On the basis of the marketing manager's prior probabilities, determine:

$($i$)$ Whether the product should be launched;

$($ii$)$ The expected value of perfect information.

b$)$ The managers have another option. Rather than going immediately for a full national

launch they could first test market the product in their Northern sales region. This

would obviously delay the national launch, and this delay, together with other outlays

associated with the test marketing, would lead to costs having a net present value of

$$3$ million. The test marketing would give an indication as to the likely success of the

national launch, and the reliability of each of the possible indications which could

result are shown by the conditional probabilities in the table below $($e$.$g$.$ if the market

for the product is such that high sales could be achieved there is a probability of $0\mathrm{.}15$

that test marketing would in fact indicate only medium sales$)$: