$1.$ A new engineer is evaluating whether to use a higher$-$voltage transmission line. It will cost $\backslash (\backslash$$ $250,000\backslash )$ more initially, but it will reduce transmission losses. The optimistic $($with probability $\backslash (20\backslash \%\backslash )),$ most likely $($with probability $\backslash (70\backslash \%\backslash )),$ and pessimistic projections for annual savings are $\backslash (\backslash$$ $20,000,\backslash$$ $15,000\backslash ),$ and $\backslash (\backslash$$ $8,000\backslash ),$ respectively. The interest rate is $\backslash (6\backslash \%\backslash ),$ and the transmission line should have a life of $30$ years.

$($a$)$ Find the probability of the pessimistic projection. $(10$ points$)$

$($b$)$ What is the present worth $($PW$)$ for each estimated value? $(10$ points$)$

$($c$)$ What is the expected value for the present worth? $(15$ points$)$

$($d$)$ What is the risk of this present worth $($use variance equation, then take the square root$)?(15$ points$)$