Question $6$

The tables below show prior, conditional, joint, marginal, and posterior probabilities for $3$ states of nature: Good, Fair, and Poor. Encouraging and Discouraging are the possible forecasts of sample information.

Encouraging

$\backslash$table$[[,$Prior,Conditional,Joint,Posterior$],[$Good$,0\mathrm{.}25,0\mathrm{.}68,0\mathrm{.}17,0\mathrm{.}323],[$Fair$,0\mathrm{.}4,0\mathrm{.}4,0\mathrm{.}16,0\mathrm{.}304],[$Poor$,0\mathrm{.}35,0\mathrm{.}56,0\mathrm{.}196,0\mathrm{.}373]]$

Discouraging

$\backslash$table$[[,$Prior,Conditional,Joint,Posterior$],[$Good$,0\mathrm{.}25,0\mathrm{.}32,0\mathrm{.}08,0\mathrm{.}169],[$Fair$,0\mathrm{.}4,0\mathrm{.}6,0\mathrm{.}24,0\mathrm{.}506],[$Poor$,0\mathrm{.}35,0\mathrm{.}44,0\mathrm{.}154,0\mathrm{.}325]]$

Use the given tables to determine the following probabilities:

a$)$ Good