https://file.aiquickdraw.com/mathbot/file/1728766131700e5mdudmz.jpg

The problem presented in the image involves calculating the expected value of a life insurance policy based on probability.

### Problem Breakdown:
- **Probability that an 80-year-old man will die within one year**: $ 0.069941 $.
- **Value of the life insurance policy**: $ \$14,000 $.
- **Premium paid by the person**: $ \$455 $.

### Solution:
The company's expectation is calculated using the concept of **expected value** (EV). The expected value formula in this case is:

\[
\text{EV} = (\text{probability of death} \times \text{payout}) + (\text{probability of survival} \times \text{premium})
\]

1. **Probability of death**: $ P(\text{death}) = 0.069941 $
2. **Probability of survival**: $ P(\text{survival}) = 1 - P(\text{death}) = 1 - 0.069941 = 0.930059 $

Now, let's apply the values to the formula:

\[
\text{EV} = (0.069941 \times -14,000) + (0.930059 \times 455)
\]

First, calculate the terms:

\[
0.069941 \times (-14,000) = -979.174
\]
\[
0.930059 \times 455 = 423.277
\]

Finally, sum these values to find the expected value:

\[
\text{EV} = -979.174 + 423.277 = -555.897
\]

Rounding to two decimal places:

\[
\boxed{-555.90}
\]

This means the insurance company's expected loss is approximately \$555.90 per policy sold to an 80-year-old man.

### Would you like more details or clarifications?

### Related Questions:
1. How would the expectation change if the insurance payout increased to \$20,000?
2. What would happen if the probability of death increased to 0.08?
3. Can you explain the relationship between risk and premium in insurance?
4. How is the expected value concept used in other fields like gambling or stock trading?
5. How would the calculation change if the premium was variable?

### Tip:
Always round the final result as instructed (e.g., to two decimal places) to avoid calculation errors in financial contexts.

The probability that an 80-year-old man in the U.S. will die within one year is approximately 0.069941. If an insurance company sells a one-year, $14,000 life insurance policy to such a person for $455, what is the company's expectation? (Round your answer to two decimal places.)

Calculate the expected value of a $14,000 life insurance policy given the probability of death is 0.069941. This insurance math problem involves concepts like expected value and probability. Discover how the company's expectation becomes negative, representing potential losses.

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