Greenfield Gardens is evaluating whether to expand its operations, maintain its current size, or reduce its

operations. The management team has assessed the demand levels for their organic produce in the local

market and assigned the following probabilities to each demand level: high $($probability $0\mathrm{.}35),$ medium

$($probability $0\mathrm{.}50),$ or low $($probability $0\mathrm{.}15).$ The expected payoffs for each demand$/$size scenario are listed

below:

Acreage

Demand Expand Same Size Contract

High $$150,000$ $$60,000$ $$30,000$

Medium $$70,000$ $$50,000$ $$30,000$

Low $$-10,700$ $$25,000$ $$30,000$

a$.$ What is perfect information about the demand worth under these circumstances?

Please show your calculations. $(4$ marks$)$

b$.$ The company has engaged an expert to forecast the demand. The expert has predicted high demand.

The accuracy of the expert's predictions is as follows:

P$($PH $$ H$)=0\mathrm{.}75,$ P$($PH $$ M$)=0\mathrm{.}15,$ P$($PH $$ L$)=0\mathrm{.}10$

P$($PM $$ H$)=0\mathrm{.}20,$ P$($PM $$ M$)=0\mathrm{.}75,$ P$($PM $$ L$)=0\mathrm{.}05$

P$($PL $$ H$)=0\mathrm{.}05,$ P$($PL $$ M$)=0\mathrm{.}10,$ P$($PL $$ L$)=0\mathrm{.}85$

Where:

H $=$ high demand PH $=$ predict high demand

M $=$ medium demand PM $=$ predict medium demand

L $=$ low demand PL $=$ predict low demand

Based on the accuracy of the prediction, calculate the revised probability for each state of nature

$($to $3$ decimal places$)$ given that the consultant forecast high demand. $(6$ marks