Question

Fire insurance Suppose a homeowner spends \(300 for a home insurance policy that will pay out \)200,000 if the home is destroyed by fire in a given year. Let P = the profit made by the company on a single policy. From previous data, the probability that a home in this area will be destroyed by fire is 0.0002.

(a) Make a table that shows the probability distribution of P.

(b) Calculate the expected value of P. Explain what this result means for the insurance company.

(c) Calculate the standard deviation of P. Explain what this result means for the insurance company.

Short answer

Part (a)

Profit	300	-$199700
Probability	0.9998	0.0002

Part (b) $260

Part (c) about 2828.14

Step-by-step solution

Part (a) Step 1. Given information.

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.

Part (a) Step 2. Make a table that shows the probability distribution of P.

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.

Profit	300	-$199700
Probability	0.9998	0.0002

Part (b) Step 1. Calculate the expected value of P.

$$\begin{gathered} \mu =\sum xP\left(X=x\right) \\ =300\times \left(0.9998\right)+\left(-199700\right)\times \left(0.0002\right) \\ =260 \end{gathered}$$

The average profit on a single policy is $260.

Part (c) Step 1. Find the standard deviation of P.

The mean is $260.

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

$$\begin{gathered} {\sigma }^2=\sum {\left(x-\mu \right)}^{2}P\left(x\right) \\ ={\left(300-260\right)}^2\times \left(0.9998\right)+{\left(-199700-260\right)}^2\times \left(0.0002\right) \\ =7,998,400 \end{gathered}$$

The standard deviation is:

$$\begin{gathered} \sigma =\sqrt{{\sigma }^2} \\ =\sqrt{7,998,400} \\ \approx 2828.14 \end{gathered}$$

We can see that on average the profit will vary from a mean of $260 to $2828.14.