Suppose a life insurance company sells a $$ \$ 300,0001 $$-year term life insurance policy to a 20 -year-old female for $$ \$ 210 $$. According to the National Vital Statistics Report, 58 ( 21 ), the probability that the female survives the year is 0999544 Compute and interpret the expected value of this policy to the insurance company. The expected value is $$ \$ 73.20 $$ (Round to the nearest cent as needed.) Which of the following interpretations of the expected value is correct? Select the correct choice below and fill in the answer box to complete your choice. (Round to the nearest cent as needed) A. The insurance company expects to make a profit of $$ \$ \square $$ on every 20 -year-old female it insures for 1 year. B. The insurance company expects to make a minimurn profit of $$ \$ \square $$ on every 20 -year-old female it insures for 1 month. O C. The insurance company expects to make a profit of $$ \$ \square $$ on every 20 -year-old female it insures for 1 month. O D. The insurance company expects to make a maximum profit of $$ \$ \square $$ on every 20 -year-old female it insures for 1 year

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To compute the expected value of the life insurance policy to the insurance company, we can use the following formula for expected value (EV):

$$
EV = (P(\text{survive}) \times \text{profit if survives}) + (P(\text{not survive}) \times \text{loss if not survives})
$$

### Known Conditions:
- The policy amount is $$ \$300,000 $$.
- The premium paid by the customer is $$ \$210 $$.
- The probability that the female survives the year is $$ P(\text{survive}) = 0.999544 $$.
- The probability that the female does not survive the year is $$ P(\text{not survive}) = 1 - P(\text{survive}) = 1 - 0.999544 = 0.000456 $$.

### Profit and Loss Calculations:
1. **If the female survives**:
   - The insurance company keeps the premium: $$ \$210 $$.
   - Profit if survives: $$ 210 $$.

2. **If the female does not survive**:
   - The insurance company pays out $$ \$300,000 $$ but has received $$ \$210 $$.
   - Loss if not survives: $$ 300,000 - 210 = 299,790 $$.

### Expected Value Calculation:
Now we can substitute these values into the expected value formula:

$$
EV = (0.999544 \times 210) + (0.000456 \times -299790)
$$

Let's calculate this step by step.

1. Calculate the profit if survives:
   $$
   0.999544 \times 210
   $$

2. Calculate the loss if not survives:
   $$
   0.000456 \times -299790
   $$

3. Add both results to find the expected value.

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$0.999544\times 210$$
- step1: Multiply the numbers:
  $$209.90424$$
Calculate or simplify the expression $$ 0.000456 * -299790 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.000456\left(-299790\right)$$
- step1: Multiply the numbers:
  $$-136.70424$$
Calculate or simplify the expression $$ 209.90424 - 136.70424 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$209.90424-136.70424$$
- step1: Subtract the numbers:
  $$73.2$$
### Expected Value Calculation Results:
1. Profit if survives:
   $$
   0.999544 \times 210 \approx 209.90
   $$

2. Loss if not survives:
   $$
   0.000456 \times -299790 \approx -136.70
   $$

3. Now, we calculate the expected value:
   $$
   EV = 209.90 - 136.70 = 73.20
   $$

### Interpretation of the Expected Value:
The expected value of $$ \$73.20 $$ means that, on average, the insurance company expects to make a profit of $$ \$73.20 $$ for each 20-year-old female it insures for one year.

### Correct Interpretation Choice:
The correct interpretation is:
**A. The insurance company expects to make a profit of $$ \$73.20 $$ on every 20-year-old female it insures for 1 year.**
The insurance company expects to make a profit of \$73.20 on every 20-year-old female it insures for 1 year.

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