Mark M$.$ Upp has just been fired as the university bookstore manager for setting prices too low$($only $20$ percent above suggested$$ retail$).$ He is considering opening a competing bookstore near the$$ campus, and he has begun an analysis of the situation. There are two possible sites under consideration. One is relatively small$($Site $1),$ while the other is large$($Site $2).$ If he opens at Site $1$ and demand is$$ high, he will generate a profit of$$ $$50,000.$ If demand is$$ low, he will lose$$ $$10,000.$ If he opens at Site $2$ and demand is$$ high, he will generate a profit of$$ $$80,000,$ but he will lose$$ $$30,000$ if demand is low. He also has the option of not opening either. He believes that there is a $50$ percent chance that demand will be high. Mark can purchase a market research for$$ $$2,000.$ The probability of high demand given favorable research results is $0\mathrm{.}8.$ The probability of high demand given unfavorable research results is $0\mathrm{.}1.$ There is a $60$ percent chance that the research results will be favorable.

Based on case$1,$ what is the prior probability of two state of$$ natures?

A$.$

$$P$($demand high$)=0\mathrm{.}5,$ P$($demand low$)=0\mathrm{.}5$

B$.$

$$P$($demand high$)=0\mathrm{.}8,$ P$($demand low$)=0\mathrm{.}2$

C$.$

$$P$($demand high$)=0\mathrm{.}6,$ P$($demand low$)=0\mathrm{.}4$

D$.$

$$P$($demand high$)=0\mathrm{.}1,$ P$($demand low$)=0\mathrm{.}9$

E$.$

None of the above