To find the expected value of each option, we need to consider both the cost and the probability of each service being needed.
### Expected Value without Insurance:
1. Annual Premium: The cost without insurance is \[tex]$0.
2. Five Doctor Visits: The probability of needing this service is 45% (or 0.45), and the cost without insurance is \$[/tex]1,750.
- Expected cost for doctor visits: [tex]\(0.45 \times 1,750 = 787.5\)[/tex]
3. Medication: The probability of needing this service is 75% (or 0.75), and the cost without insurance is \[tex]$325.
- Expected cost for medication: \(0.75 \times 325 = 243.75\)
Adding these together gives the total expected value without insurance:
\[ 787.5 + 243.75 = 1031.25 \]
### Expected Value with Insurance:
1. Annual Premium: The cost with insurance is \$[/tex]1,580.
2. Five Doctor Visits: The probability of needing this service is 45% (or 0.45), and the cost with insurance is \[tex]$125.
- Expected cost for doctor visits: \(0.45 \times 125 = 56.25\)
3. Medication: The probability of needing this service is 75% (or 0.75), and the cost with insurance is \$[/tex]75.
- Expected cost for medication: [tex]\(0.75 \times 75 = 56.25\)[/tex]
Adding these together gives the total expected value with insurance:
[tex]\[ 1,580 + 56.25 + 56.25 = 1692.5 \][/tex]
Therefore:
- The expected value of health care without insurance is [tex]$\$[/tex]1031.25[tex]$
- The expected value of health care with insurance is $[/tex]\[tex]$1692.5$[/tex]
