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

$($i$)$ Whether the product should be launched$(8)$ 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 is 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$)$:

Test marketing indication

Actual national

sales

High

Medium

Medium

Medium national

sales

sales

Low national sales

High national sales

$0\mathrm{.}80$

$0\mathrm{.}15$

$0\mathrm{.}05$

$0\mathrm{.}60$

$0\mathrm{.}15$

Low

$0\mathrm{.}10$

$0\mathrm{.}30$

$0\mathrm{.}60$

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