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

$\backslash$table$[[$Payment $($$$),$Probability$],[0,0\mathrm{.}81],[500,0\mathrm{.}08],[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 annual insurance premium $($in dollars$)$ that would enable the company to break even.

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$($b$)$ The insurance company charges an annual premium of $$540$ 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.$)$

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Why does the policy holder purchase a collision policy with this expected value?

The expected value of the policy is zero indicating on average buying the policy results in a gain for the policy holder.

The expected value of the policy is negative indicating on average buying the policy results in a loss for the policy holder. However, the policy protects the policy holder against a large repair bill in the event of an accident.

The expected value of the policy is positive indicating on average buying the policy results in a gain for the policy holder.

The expected value of the policy is positive indicating on average buying the policy results in a loss for the policy holder. However, the policy protects the policy holder against a large repair bill in the event of an accident.

The expected value of the policy is negative indicating on average buying the pollcy results in a gain for the pollcy holder.