Question

An insurance company sells a 1-year term life insurance policy to an 79-year-old woman. The woman pays a premium of $1700. If she dies within 1 year, the company will pay $43,000 to her beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 79-year-old woman will be alive 1 year later is 0.9605. Let X be the profit made by the insurance company.

(a) Find the probability distribution.

The probability distribution is x| [?] 1700
P(x)| [?] [?]

Answer 1

To find the probability distribution for the profit \( X \) made by the insurance company, we need to consider the two possible outcomes: the woman survives the year and the woman dies within the year.

1. **If the woman survives (with a probability of 0.9605):**
- The insurance company keeps her premium of $1700 and pays out nothing.
- Profit \( X \) = Premium - Payout = $1700 - $0 = $1700.

2. **If the woman dies (with a probability of \( 1 - 0.9605 = 0.0395 \)):**
- The insurance company collects the premium but must pay out $43,000 to her beneficiary.
- Profit \( X \) = Premium - Payout = $1700 - $43000 = -$41300.

Now, we can summarize the profit outcomes and their probabilities:

- Probability distribution:

\[
\begin{array}{|c|c|}
\hline
X & P(X) \\
\hline
1700 & 0.9605 \\
-41300 & 0.0395 \\
\hline
\end{array}
\]

Thus, the final probability distribution is:

- \( x \) : 1700 and -41300
- \( P(x) \) : 0.9605 and 0.0395

Answer 2

(b) Find the expected value of the profit.

Answer 3

To find the expected value \( E(X) \) of the profit \( X \), we use the formula for expected value:

\[
E(X) = \sum [x \cdot P(x)]
\]

Using the probability distribution we found in part (a), the values of \( x \) and their corresponding probabilities \( P(x) \) are:

1. \( X = 1700 \) with \( P(X = 1700) = 0.9605 \)
2. \( X = -41300 \) with \( P(X = -41300) = 0.0395 \)

Now, we can calculate the expected value:

\[
E(X) = (1700 \cdot 0.9605) + (-41300 \cdot 0.0395)
\]

Calculating each term:

1. \( 1700 \cdot 0.9605 = 1633.85 \)
2. \( -41300 \cdot 0.0395 = -1628.85 \)

Now, we add the two results:

\[
E(X) = 1633.85 - 1628.85 = 5
\]

Thus, the expected value of the profit \( E(X) \) is:

\[
\boxed{5}
\]