Suppose a life insurance company sells a $$ \$ 170,0001 $$-year term life insurance policy to a 20 -year-old female for $$ \$ 190 $$. According to the National Vital $$ S t a t i s t i c s $$ Report, 58 (21). the probability that the female survives the year is 0.999544 . Compute and interpret the expected value of this policy to the insurance company. The expected value is $$ \$ \square $$.

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To compute the expected value of the life insurance policy to the insurance company, we'll consider the two possible outcomes:

1. **Policyholder Survives the Year**:
   - **Probability**: $$ 0.999544 $$
   - **Outcome**: The insurance company keeps the premium of \$190.
   - **Net Profit**: \$190

2. **Policyholder Passes Away within the Year**:
   - **Probability**: $$ 1 - 0.999544 = 0.000456 $$
   - **Outcome**: The insurance company pays out \$170,000 but has already collected \$190.
   - **Net Loss**: $$ \$170,000 - \$190 = \$169,810 $$

Now, calculate the expected value (E) by considering both scenarios:

$$
E = (\text{Probability of Survival} \times \text{Profit if Survives}) + (\text{Probability of Death} \times \text{Loss if Dies})
$$

$$
E = (0.999544 \times 190) + (0.000456 \times (-169,810))
$$

$$
E = 189.91336 - 77.43336 = 112.48
$$

**Interpretation**:
The expected value of the policy to the insurance company is **\$112.48**. This means that, on average, the insurance company expects to earn \$112.48 from selling this policy to a 20-year-old female.

**Answer**:
The expected value is \$112.48.
The expected value of the policy is \$112.48.

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