Example $2$: Dan is pricing car insurance from a new company that only pays out at $3$ levels: total, major accident and fender bender. His car is valued at $$30,000.$ After reviewing his application, his insurance company assigned him to a group with the probabilities listed below.

These values are used to figure the value the company must charge to cover their payouts. If the company charges a $$250$ service fee, what will be his annual premium?

a$)$ Find the value for x in the P$($x$)$ column.

b$)$ Find the expected value of the company payout using $=\backslash$Sigma $(()).$

c$)$ To figure Dan$$s premium the insurance company uses his expected value amount and adds a service fee of $$250.$ Find the value.

d$)$ Does this value seem reasonable for a New Jersey driver?