Let's break down the problem step by step to find the expected value of the policy for the insurance company.
### Step-by-Step Solution:
1. Identify the variables:
- Policy value: \[tex]$275,336
- Annual cost (premium): \$[/tex]243
- Probability of death: 0.0005
2. Calculate the expected payout by the insurance company:
The expected payout is calculated by multiplying the policy value by the probability of death:
[tex]\[
\text{Expected payout} = \text{Policy value} \times \text{Probability of death}
\][/tex]
Substituting the given values:
[tex]\[
\text{Expected payout} = \$275,336 \times 0.0005 = \$137.668
\][/tex]
3. Calculate the expected profit for the insurance company:
The expected profit is calculated by subtracting the expected payout from the annual cost:
[tex]\[
\text{Expected profit} = \text{Annual cost} - \text{Expected payout}
\][/tex]
Substituting the given values:
[tex]\[
\text{Expected profit} = \$243 - \$137.668 = \$105.332
\][/tex]
### Conclusion:
The expected value of the policy for the insurance company is comprised of both the expected payout and the expected profit:
- Expected payout: \[tex]$137.668
- Expected profit: \$[/tex]105.332
In other words, the insurance company can expect, on average, to pay out \[tex]$137.668 but still retain a profit of \$[/tex]105.332 from the premium collected.
