Farm Grown, Inc., produces cases of perishable food products. Each case contains an assortment of vegetables and other farm products. Each case costs $$20$ and sells for $$45.$ If there are any cases not sold by the end of the day, they are sold to a large food processing company for $$5$ a case. There are $3$ decision alternatives. Alternative $1$: produce $100$ cases; Alternative $2$: produce $200$ cases; and Alternative $3$: produce $300$ cases. The probability that daily demand will be $100$ cases $($low demand$)$ is $0\mathrm{.}2,$ the probability that daily demand will be $200$ cases $($medium demand$)$ is $0\mathrm{.}4,$ and the probability that daily demand will be $300$ cases $($high demand$)$ is $0\mathrm{.}4.$ Farm Grown has a policy of always satisfying customer demands. If its own supply of cases is less than the demand, it buys the necessary vegetables from a competitor. The estimated cost of doing this is $$40$ per case.

Farm Grown is also considering hiring a business student at a cost of $$300$ to get additional information on the future demand. The additional information can be positive or negative. Positive information would mean that a high demand is likely and negative information would mean that a low demand is likely.

Given the following conditional probabilities:

P$($Positive Information $|$ High demand$)=0\mathrm{.}80$

P$($Positive Information $|$ Medium demand$)=0\mathrm{.}50$

P$($Negative Information $|$ Low demand$)=0\mathrm{.}72$

Calculate the following posterior probabilities: $[2]$

P$($High demand $|$ Positive Information$)$

P$($Medium demand $|$ Negative Information$)$

P$($Low demand $|$ Negative Information$)$

$($Keep answers to four decimal places$)$

simple and in a presentation style