Suppose a life insurance company selts a \( \$ 300,000 \) i-year term life insurance policy to a 20-year-old fomale for \( \$ 210 \). According to the National Vital Statistics Report, \( 58(21 \) ), the probability that the fomale survives the year is 0.999544 . Compute and interpret the expected value of this policy to the insurance company. The expected value is \( \$ \square \) (Round to the nearest cent as needed.)

To calculate the expected value of the life insurance policy for the insurance company, we will consider the two possible outcomes: 1. The female survives the year. 2. The female does not survive the year. Let’s denote: - \( P(\text{Survives}) = 0.999544 \) - \( P(\text{Does not survive}) = 1 - P(\text{Survives}) = 0.000456 \) If she survives, the insurance company collects the premium of \( \$210 \). If she does not survive, the company has to pay out \( \$300,000 \) minus the premium received, resulting in a net loss of \( \$299,790 \). Now, we can compute the expected value \( EV \): \[ EV = (P(\text{Survives}) \times \text{Gain if survives}) + (P(\text{Does not survive}) \times \text{Loss if not survives}) \] Substituting the values: \[ EV = (0.999544 \times 210) + (0.000456 \times (-299790)) \] Calculating: 1. Gain if survives: \[ 0.999544 \times 210 = 209.88 \] 2. Loss if does not survive: \[ 0.000456 \times (-299790) \approx -136.50 \] Adding these together: \[ EV = 209.88 - 136.50 = 73.38 \] Thus, the expected value of the policy to the insurance company is approximately \( \$73.38 \). The expected value is \( \$ 73.38 \).