Construct a decision tree and calculate the probabilities and EMV for each decision using the information given. Your company has identified and purchased a plot of land suitable to develop a small lithium mine. Lithium is in high demand due to its use in lightweight batteries, including for electric vehicles. Lithium is found in three kinds of deposits, which we will call type $1,$ type $2,$ and type $3.$ While you are sure that the site contains lithium, you do not know for certain what kind of deposit it holds. Based on the geological work you have already done, you estimate a $40\%$ chance of a type $1$ deposit, a $35\%$ chance of a type $2$ deposit, and a $25\%$ chance of a type $3$ deposit.

There are two kinds of mines you could build on the site, a $$bore mine$$ or a $$pit mine$.$ The payoffs in millions of dollars for each kind of mine and deposit are as follows:Type of Deposit

Type $1$ Type $2$ Type $3$

Bore mine $905020$

Pit mine $508515$

For example, if construct a bore mine and the deposit turns out to be of type $1,$ your profit would be $$90$ million.

You also have standing offer from a competing company to buy the site for $$40$ million, which you may accept instead of developing the site.

Before making any other decisions, you have the option of performing a seismic reflection study of the site, which would be more detailed than your prior geological work and cost $$2$ million. This study will predict the type of the deposit to be of type $1,2,$ or $3.$ The vendor that would perform the seismic reflection study has provided the following historical data:.

Seismic prediction probabilities conditional on deposit type

Type $1$ Type $2$ Type $3$

Predict $10\mathrm{.}751/70\mathrm{.}28$

Predict $20\mathrm{.}155/70\mathrm{.}20$

Predict $30\mathrm{.}101/70\mathrm{.}52$

That is$,$ for what turned out to be type $1$ deposits, seismic reflection studies have made the correct prediction in $75\%$ of past cases, have $($incorrectly$)$ predicted type $2$ in $15\%$ of past cases, and $($also incorrectly$)$ predicted type $3$ in $10\%$ of past cases. For what turned out to be type $2$ deposits, past reflection seismic studies have correctly predicted a type $2$ deposit in $5$ out of $7$ cases, incorrectly predicted a type $1$ deposit in $1$ out of $7$ cases, and incorrectly predicted a type $3$ deposit in $1$ out of $7$ cases. The $$type $3$ column of the table is interpreted similarly.

If you perform the seismic reflection study, you may still accept the offer to sell the site to your competitor after getting the study results, no matter what they turn out to be$.$

Using a Bayes analysis and a decision tree, find course of action that optimizes your EMV in this situation. State the optimal policy and its EMV.

What is the EVSI of the seismic reflection survey?

What is the EVPI in this situation?