Question

Fire insurance Suppose a homeowner spends $\backslash (300$for a home insurance policy that will pay out $\backslash )200,000$if the home is destroyed by fire. Let $Y=$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 Y.

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

Short answer

1. The probability distribution of $Y$is

$\begin{array}{|lrr|}\hline \mathbf{Y}(\$)& 199700& 300\\ \text{Probability}& 0.0002& 0.9998\\ \hline\end{array}$

b. The expected value of$Y$ is$\$260.$

Step-by-step solution

Part (a) Step 1: Given Information

Given in the question that,

Amount spend on insurance $=\$300$

Amount insurance will pay if fire destroys home $=\$200000$

Probability that fire will destroy the home$=0.0002$

Part (a) Step 2: Calculation 

Using the given information, the probability distribution is:

$\begin{array}{|lrr|}\hline \mathrm{Y}& 300-200000& 300\\ \text{Probability}& 0.0002& 1-0.0002\\ \hline\end{array}$

The probability distribution is

$\begin{array}{|lrr|}\hline \mathbf{Y}(\$)& 199700& 300\\ \text{Probability}& 0.0002& 0.9998\\ \hline\end{array}$

Part (b) Step 1: Given Information

Given in the question that,

Amount spend on insurance$=\$300$

Amount insurance will pay if fire destroys home $=\$200000$

Probability that fire will destroy the home$=0.0002$

Part (b) Step 2: Calculation 

The expected value can be computed using the following formula:

$$\begin{gathered} \mathrm{E}\left(Y\right)=\sum y\times P\left(y\right) \\ =300\left(0.9998\right)+(-199700)\left(0.0002\right) \\ =260 \end{gathered}$$

The insurance company's estimated profit per homeowner is$\$260$.