Question $5.(10$ points$)$ A carnival has a dart game consisting of three circles with different payoffs as:

$($Nothing is won if the player

misses the dart board$)$

Over time the operator has learned that when a customer throws a dart, they hit each region with the fol$-$

lowing probabilities: $p(A)=0\mathrm{.}02,p(B)=0\mathrm{.}06,$ and $p(C)=0\mathrm{.}1.$

a$)$ What is the expected payoff for one throw of a dart?

b$)$ If a customer pays $$5$ to throw $3$ darts, how much can the carnival expect to make per customer? $($If a

customer throws all three darts in A they win $$20+$$$20+$$$20=$$$60,$ etc....$)$