Question

An insurance company sells a 1-year term life insurance policy to an 81-year-old woman. The woman pays a premium of 2700. If she dies within 1 year, the company will pay 92,000 to her beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 81-year-old woman will be alive year later is . Let X be the profit made by the insurance company.
Find the probability distribution.

Answer 1

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

1. **Define the associated probabilities:**
- Let \( p \) be the probability that the 81-year-old woman will die within the year.
- The probability that she will be alive after 1 year is then \( 1 - p \).

According to the question, we don't have the exact value of \( p \) from the CDC. However, we can denote the probability of survival as \( p_{\text{alive}} \). Thus, \( p = 1 - p_{\text{alive}} \).

2. **Profit outcomes:**
- If the woman **lives** for the year:
- The insurance company keeps the premium of \( 2700 \) since no payout is made.
- Profit \( X = 2700 \).

- If the woman **dies** within the year:
- The insurance company has to pay out \( 92,000 \).
- Profit \( X = 2700 - 92000 = -89200 \).

3. **Construct the probability distribution:**
Let’s define the outcomes and their associated probabilities:

- \( X = 2700 \) with probability \( p_{\text{alive}} \)
- \( X = -89200 \) with probability \( p_{\text{die}} = 1 - p_{\text{alive}} \)

4. **Summary of the probability distribution:**
We can summarize this as follows:

\[
\begin{array}{|c|c|c|}
\hline
X & \text{Probability} \\
\hline
2700 & p_{\text{alive}} \\
-89200 & 1 - p_{\text{alive}} \\
\hline
\end{array}
\]

### Conclusion
The profit \( X \) has a discrete probability distribution characterized with \( 2700 \) with a probability of survival (denote as \( p_{\text{alive}} \)) and \(-89200\) with a probability of dying (denote as \( 1 - p_{\text{alive}} \)). To fully determine this distribution, you would substitute \( p_{\text{alive}} \) with the actual probability from the CDC's data for an 81-year-old woman's survival rate.

Answer 2

the probability is 0.9714
now compute

Answer 3

Given that the probability that an 81-year-old woman will be alive one year later is \( p_{\text{alive}} = 0.9714 \), we can calculate the probabilities for the profit outcomes.

1. **Calculate the probability of dying:**
\[
p_{\text{die}} = 1 - p_{\text{alive}} = 1 - 0.9714 = 0.0286.
\]

2. **Profit outcomes:**
- If the woman **lives** (probability \( p_{\text{alive}} = 0.9714 \)):
- Profit \( X = 2700 \)

- If the woman **dies** (probability \( p_{\text{die}} = 0.0286 \)):
- Profit \( X = -89200 \)

3. **Summary of the probability distribution:**
\[
\begin{array}{|c|c|c|}
\hline
X & \text{Probability} \\
\hline
2700 & 0.9714 \\
-89200 & 0.0286 \\
\hline
\end{array}
\]

### Final Result
- Profit \( X \):
- \( 2700 \) with probability \( 0.9714 \)
- \( -89200 \) with probability \( 0.0286 \)

### Expected Profit Calculation
To further examine the profitability, we can calculate the expected profit \( E(X) \):

\[
E(X) = (2700 \times 0.9714) + (-89200 \times 0.0286)
\]

Calculating the values:

\[
E(X) = (2700 \times 0.9714) + (-89200 \times 0.0286)
\]
\[
= 2622.78 - 2545.92
\]
\[
= 76.86
\]

### Conclusion
The expected profit \( E(X) \) for the insurance company from this policy is approximately **$76.86**. The insurance company can expect to make a profit of roughly $76.86 on average for each policy sold to an 81-year-old woman with these parameters.

The probability distribution has been established, along with the expected value of the profit from the policy.

Answer 4

its not -89,200 its 89,300. Now compute the new expected value of the profit

Answer 5

Thank you for the correction regarding the profit related to the woman's death. Let's update the calculations accordingly, using the correct value of \( 89,300 \) for the payout.

### Updated Profit Outcomes
1. **If the woman lives** (probability \( p_{\text{alive}} = 0.9714 \)):
- Profit \( X = 2700 \)

2. **If the woman dies** (probability \( p_{\text{die}} = 0.0286 \)):
- Profit \( X = 2700 - 89300 = -86600 \)

### Summary of the Probability Distribution:
\[
\begin{array}{|c|c|c|}
\hline
X & \text{Probability} \\
\hline
2700 & 0.9714 \\
-86600 & 0.0286 \\
\hline
\end{array}
\]

### Expected Profit Calculation
To calculate the expected profit \( E(X) \):

\[
E(X) = (2700 \times 0.9714) + (-86600 \times 0.0286)
\]

Calculating each term:

1. **Live Outcome:**
\[
2700 \times 0.9714 = 2622.78
\]

2. **Die Outcome:**
\[
-86600 \times 0.0286 = -24751.80
\]

Now, sum these to find the expected profit:

\[
E(X) = 2622.78 - 24751.80
\]
\[
E(X) = -22129.02
\]

### Conclusion
The expected profit \( E(X) \) for the insurance company from this policy is approximately **\(-22,129.02\)**. This indicates that, on average, the insurance company would incur a loss of about **$22,129.02** for each policy sold under these conditions.

Answer 6

b) Find the expected value of the profit.

Answer 7

To find the expected value of the profit from the life insurance policy, let's outline the outcomes again considering the corrected payout.

### Outcomes and Probabilities
1. **If the woman lives** (probability \( p_{\text{alive}} = 0.9714 \)):
- Profit \( X = 2700 \)

2. **If the woman dies** (probability \( p_{\text{die}} = 0.0286 \)):
- Profit \( X = 2700 - 89300 = -86600 \)

### Summary of the Probability Distribution
\[
\begin{array}{|c|c|c|}
\hline
X & \text{Probability} \\
\hline
2700 & 0.9714 \\
-86600 & 0.0286 \\
\hline
\end{array}
\]

### Expected Value Calculation
To compute the expected value \( E(X) \):
\[
E(X) = (2700 \times 0.9714) + (-86600 \times 0.0286)
\]

Let's calculate each term:

1. **Live Outcome:**
\[
2700 \times 0.9714 = 2622.78
\]

2. **Die Outcome:**
\[
-86600 \times 0.0286 = -24751.80
\]

### Combining the Results
Now, we can combine these results to find the expected profit:
\[
E(X) = 2622.78 - 24751.80
\]
\[
E(X) = -22129.02
\]

### Conclusion
The expected value of the profit \( E(X) \) from selling this life insurance policy is approximately **\(-22,129.02\)**. This suggests that, on average, the insurance company would incur a loss of about **$22,129.02** for each policy sold under these parameters.