$```$

Question $5(10$ marks$)$:

$```$

The Goferbroke Company develops oil wells in unproven territory. A consulting geologist has reported that there is a $\backslash (28\backslash \%\backslash )$ of oil on a particular tract of land. Drilling for oil on this tract would require an investment of about $\backslash (\backslash$$ $200,000\backslash ).$ If the tract contains oil, it is estimated that the net revenue generated would be approximately $\backslash (\backslash$$ $800,000\backslash ).$ Another oil company has offered to purchase the tract of land for $\backslash$$$90,000.$

Goferbroke can obtain improved estimates of the chance of oil by conducting a detailed seismic survey of the land. Possible findings from a seismic survey:

F: Favorable scismic soundings; oil is fairly likely.

U : Unfavorable seismic soundings; oil is quite unlikely.

Conditional Probabilities

State of

Nature

Favorable $($F$)$ Lnfavorable$($U$)$

Oil $($S$1)$

$\backslash ($ R$(\backslash$mathrm$\{$~F$\}\backslash$mid $\backslash$mathrm$\{$S$\}1)=0\mathrm{.}29\backslash )$

$\backslash ($ P$(\backslash$mathcal$\{$U$\}\backslash$mid $\backslash$mathrm$\{$S$\}1)=0\mathrm{.}71\backslash )$

Dry $($S$2)$

$\backslash ($ R$(\backslash$mathrm$\{$~F$\}\backslash$mid $\backslash$mathrm$\{$S$\}2)=0\mathrm{.}13\backslash )$

$\backslash ($ R$(\backslash$mathrm$\{$U$\}\backslash$mid $\backslash$mathrm$\{$S$\}2)=0\mathrm{.}87\backslash )$

Calculate the posterior probabilities in the following tables $(4$ marks$)$

$($Round your answers to $3$ decimal places.$)$

B$($SjLb$)$

P$($Sidu$)$

Incorporating the prior and posterior probabilities in the given decision tree for Goferbrake Company here: