The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance follows.

$\backslash$table$[[$Payment $($$$),$Probability$],[0,0\mathrm{.}85],[500,0\mathrm{.}04],[1,000,0\mathrm{.}04],[3,000,0\mathrm{.}03],[5,000,0\mathrm{.}02],[8,000,0\mathrm{.}01],[10,000,0\mathrm{.}01]]$

$($a$)$ Use the expected collision payment to determine the collision insurance premium $($in dollars$)$ that would enable the company to break even. $

$($b$)$ The insurance company charges an annual rate of $$580$ for the collision coverage. What is the expected value of the collision policy $($in dollars$)$ for a policyholder? $($Hint: It is the expected payments from the company minus the cost of coverage.$)$ Why does the policy holder purchase a collision policy with this expected value?

expected value of the policy $=$$