91Ó°ÊÓ

The given information is:

A homeowner pays $300 for a home insurance policy that pays out $200,000 if the house burns down in a given year.

Pis the profit made on a single policy by the company.

A home in this neighborhood has a 0.0002 chance of being destroyed by fire.

If the house is not destroyed by fire, the profit on this insurance is 300 with a chance of 0.9998 (calculated using the complement rule) because the premium was paid but the face value did not have to be paid by the firm.

$$\begin{gathered} \mathrm{P}(\mathrm{House}\mathrm{not}\mathrm{destroyed})=1-\mathrm{P}(\mathrm{House}\mathrm{destroyed}) \\ =1-0.0002 \\ =0.9998 \end{gathered}$$

Since the premium of 300 was received and the face value of 200,000 had to be paid out, the profit is $\$300-\$200,000=-\$199,700$with a chance of 0.0002, assuming the house is destroyed.

$$\begin{gathered} \mathrm{μ}=∑xP\left(X=x\right) \\ =300×\left(0.9998\right)+\left(-199700\right)×\left(0.0002\right) \\ =260 \end{gathered}$$

The average profit on a single policy is $260.

The mean is $260.

The predicted value of the squared departure from the mean is known as the variance:

$$\begin{gathered} {\mathrm{σ}}^2=∑{\left(x-\mathrm{μ}\right)}^{2}P\left(x\right) \\ ={\left(300-260\right)}^2×\left(0.9998\right)+{\left(-199700-260\right)}^2×\left(0.0002\right) \\ =7,998,400 \end{gathered}$$

The standard deviation is:

$$\begin{gathered} \mathrm{σ}=\sqrt{{\mathrm{σ}}^2} \\ =\sqrt{7,998,400} \\ ≈2828.14 \end{gathered}$$