A Manufacturing Company Introduces Two Product Alternatives ​
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the Probabilities for the State of Nature Are

A Manufacturing company introduces two product alternatives. The table below provides profit payoffs in thousands of dollars.
​ $$\begin{array} { | c | c | c | c | } \hline { \text { Bet on } } & & { \text { State of Nature (Demand) } } \\\hline & \mathrm { Up } & \text { Stable } & \text { Down } \\\hline \text { Product A } & 11 & 8 & 8 \\\hline \text { Product B } & 8 & 10 & 12 \\\hline\end{array}$$ ​
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The probabilities for the state of nature are P(Up) = 0.35, P(Stable) = 0.35, and P(Down) = 0.30.
A test market study of the potential demand for the product is expected to report either a favorable (F) or unfavorable (U) condition. The relevant conditional probabilities are as follows:
P(F|Up) = 0.5; P(F|Stable) = 0.3; P(F|Down) = 0.2
P(U|Up) = 0.2; P(U|Stable) = 0.3; P(U|Down) = 0.5
Use Bayes' theorem to compute the conditional probability of the demand being up, stable, or down, given each market research outcome.