Question $5(10$ marks$)$:

$$

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

$$

$$ Payoff Table $($Profit in $Thousands$)$

State of Nature

Alternative

$($S$1=$Oil$)$

$($S$2=$Dry$)$

Drill for oil $($d$1)$

$600$

$200$

Sell the land $($d$2)$

$90$

$90$

Prior probability

$0\mathrm{.}28$

$0\mathrm{.}72$

$$

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 seismic soundings; oil is fairly likely.

U: Unfavorable seismic soundings; oil is quite unlikely.

$$

Conditional Probabilities

State of Nature

Favorable $($F$)$

Unfavorable$($U$)$

Oil $($S$1)$

P$($F $|$ S$1)=0\mathrm{.}29$

P$($U $|$ S$1)=0\mathrm{.}71$

Dry $($S$2)$

P$($F $|$ S$2)=0\mathrm{.}13$

P$($U$|$ S$2)=0\mathrm{.}87$

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

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

P$($Sj$$F$)$

Sj

P$($Sj$)$

P$($F$$Sj$)$

P$($Sj$)$ P$($F$$Sj$)$

P$($Sj$$F$)$

S$1$

$0\mathrm{.}28$

$$

S$2$

$0\mathrm{.}72$

$$

$1\mathrm{.}00$

$$

$$

P$($Sj$$U$)$

Sj

P$($Sj$)$

P$($U$$Sj$)$

P$($Sj$)$ P$($U$$Sj$)$

P$($Sj$$U$)$

S$1$

$0\mathrm{.}28$

$$

$$

S$2$

$0\mathrm{.}72$

$$

$$

$$

$1\mathrm{.}00$

$$

$$

$$

$$

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

$$

Compute the expected values $($EVs$)$ in the following table $(5$ marks$)$::

$$

EV$($Node$)$

Values

EV$(1)=$ EVSI

$$

EV$(2)=$ EVwSI

$$

EV$(3)=$ EVwoSI

$$

EV$(4)$

$$

EV$(5)$

$$

$$

If the seismic survey of the land costs $$10,000$ determine whether the company should do the survey. If it does, what is the optimal decision strategy $(1$ mark$)$: