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

Payment $($$$)$ Probability

$00\mathrm{.}84$

$5000\mathrm{.}05$

$1,0000\mathrm{.}04$

$3,0000\mathrm{.}03$

$5,0000\mathrm{.}02$

$8,0000\mathrm{.}01$

$10,0000\mathrm{.}01$

$($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 $$525$ 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?

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

B$.$ 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.

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

D$.$ 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.

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