A national grocer is considering opening a new superstore in a rapidly growing town. The town has a few small grocery stores, but recent population increases suggest there might now be sufficient demand to support a superstore. Using a $10-$year planning horizon, the grocer has calculated the potential net present value $($NPV$)$ of the superstore under three different demand scenarios: low, medium, and high demand. The grocer's market analysts have also calculated the prior probabilities of low, medium, and high demand.

$\backslash$table$[[$Demand$,$Prior probability,NPV$],[$Low$,0\mathrm{.}4,-5$ million$],[$Medium$,0\mathrm{.}4,2$ million$],[$High$,0\mathrm{.}2,7$ million$]]$

The grocer has been contacted by a local economic planning consultancy, who have offered to conduct a major survey of the town's population to estimate the number of potential customers for the superstore. However, the cost of the survey has yet to be agreed.

The consultancy has performed similar surveys in the past for different retailers, and has provided the following data on their survey reliabilities, represented by conditional probabilities:

$\backslash$table$[[$Actual demand $($down$),$Survey indication $($across$)],[,$Low,Medium,High,$],[$Low$,0\mathrm{.}6,0\mathrm{.}2,0\mathrm{.}2,],[$Medium$,0\mathrm{.}3,0\mathrm{.}5,0\mathrm{.}2,],[$High$,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}5,]]$

Examples

If the actual demand is low, then the probability of the survey correctly indicating that demand is low too is $0\mathrm{.}6,$ or $60\%.$

If the actual demand is low, then the probability of the survey incorrectly indicating that demand is medium is $0\mathrm{.}2,$ or $20\%.$

If the actual demand is low, then the probability of the survey incorrectly indicating that demand is high is $0\mathrm{.}2,$ or $20\%.$

Part A

What is the expected value of the information provided by the local economic consultancy's survey?

$[$Question $2,$ Part A: $10$ marks$]$

Part B

Assume that the local economic consultancy's survey is perfectly reliable, e$.$g$.$ if the actual demand is low, then the probability of the survey correctly indicating that demand is low too is $1\mathrm{.}0,$ or $100\%.$

What is the expected value of the perfect information?