Life Insurance A young adult purchases a $1-$year life insurance policy worth $$250,000.$ Using historical data, the insurance company determines that the person will survive the policy period year with a probability of $0\mathrm{.}9986.$

$($a$)$ If the price of the policy is $$450,$ find the expected profit for the insurance company.

$($Hint: if the person survives the policy period, the insurance company's profit is $$450,$ and if the person does not survive the policy period, the insurance company's profit $($actually loss$)$ is $$450-$$$250,000$

$($b$)$ What price $P$ should the insurance company charge for the same policy if the insurance company wishes to have an expected profit of $$250$ for the policy?

$($Hint: if the person survives the policy period, the insurance company's profit is $$P,$ and if the person does not survive the policy period, the insurance company's profit $($actually loss$)$ is

$$p-$$$250,000$

Find the expected profit in terms of $P,$ and set it equal to $$250,$ and solve the equation for $P)$