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QUESTION

An oil company is considering an offshore drilling venture. A
preliminary survey indicates that there may be large ($L$) or
small ($S$) deposits with probabilities ${\rm P}(L) = 0.1$ and
${\rm P}(S) = 0.9$.  Exploration costs \pounds$1\,000\,000$.  If
large deposits are found, the company makes a profit of
\pounds$16\,000\,000$, less the exploration cost.  If small
deposits are discovered, the project will be abandoned.

A team of experts can be employed for \pounds$500\,000$ to perform
a survey.  They would find that either large deposits are likely
($P$), large deposits are unlikely ($N$), or there is inconclusive
evidence ($I$).  The previous record of experts is shown in the
following table.

\begin{center}
\begin{tabular}{cccc}
\hline &\multicolumn{3}{l} {Prediction}\\ \cline{2-4} Actual
result &$P$&$N$&$I$\\ \hline L&60\%&20\%&20\%\\ S&10\%&80\%&10\%
\\ \hline
\end{tabular}
\end{center}

\begin{description}
\item[(a)] Determine the probabilities of the predictions ${\rm P}(P)$,
${\rm P}(N)$ and ${\rm P}(I)$.
\item[(b)] Determine the posterior probabilities for each possible
prediction.
\item[(c)] Use a decision tree to find what course of action the company
should follow.
\end{description}

\bigskip

ANSWER


The table gives

\begin{tabular}{ccc}
P(P|L)=0.6&P(N|L)=0.2&P(I|L)=0.2\\
P(P|S)=0.1&P(N|S)=0.8&P(I|S)=0.1
\end{tabular}

\begin{description}

\item[(a)]

\begin{eqnarray*}
P(P)&=&P(P|L)P(L)+P(P|S)P(S)=(0.6\times0.1)+(0.1\times0.9)=0.15\\
P(N)&=&P(N|L)P(L)+P(N|S)P(S)=(0.2\times0.1)+(0.8\times0.9)=0.74\\
P(I)&=&P(I|L)P(L)+P(I|S)P(S)=(0.2\times0.1)+(0.1\times0.9)=0.11
\end{eqnarray*}

\item[(b)]

\begin{eqnarray*}
P(L|P)&=&\frac{P(P\cap L)}{P(P)}=\frac{P(P|L)P(L)}{P(P)}=0.4\\
P(S|P)&=&\frac{P(P\cap S)}{P(P)}=\frac{P(P|S)P(S)}{P(P)}=0.6\\
P(L|N)&=&\frac{P(N\cap L)}{P(N)}+\frac{P(N|L)P(L)}{P(N)}=0.027\\
P(S|N)&=&\frac{P(N\cap S)}{P(N)}=\frac{P(N|S)P(S)}{P(N)}=0.973\\
P(L|I)&+&\frac{P(I\cap L)}{P(I)}=\frac{P(I|L)P(L)}{P(I)}=0.182\\
P(S|I)&=&\frac{P(I\cap S)}{P(I)}=\frac{P(I|S)P(S)}{P(I)}=0.818
\end{eqnarray*}

Returns (without survey) are:

\begin{tabular}{c|cc}
&Action\\ &$E$&$\overline{E}$\\ \hline L&15&0\\ S&-1&0
\end{tabular}
$E$=explore, $\overline{E}$=do not explore


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$P(L)=0.1,\ P(S)=0.9$ are prior probabilities.

From the table:

 $P(P|L)=0.6,\ P(N|L)=0.2\ P(I|L)=0.2,\\
P(P|S)=0.1,\ P(N|S)=0.8,\ P(I|S)=0.1$


\end{description}



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