To find the expected value of health care without insurance, we will use the provided data and the formula for expected value which takes into account the probability of an event happening and the cost associated with that event.
### Without Insurance
1. Annual Premium: [tex]$0$[/tex]
2. Doctor Visits:
- Probability of needing service: [tex]$24\%$[/tex] or [tex]$0.24$[/tex]
- Cost without insurance: [tex]$\$[/tex]1400[tex]$
- Expected cost for doctor visits: $[/tex]0.24 \times 1400 = \[tex]$336$[/tex]
3. Medication:
- Probability of needing service: [tex]$12\%$[/tex] or [tex]$0.12$[/tex]
- Cost without insurance: [tex]$\$[/tex]240[tex]$
- Expected cost for medication: $[/tex]0.12 \times 240 = \[tex]$28.8$[/tex]
Sum of all expected costs without insurance:
- Expected value without insurance: [tex]$0 + 336 + 28.8 = \$[/tex]364.8[tex]$
### With Insurance
1. Annual Premium: $[/tex]\[tex]$1580$[/tex]
2. Doctor Visits:
- Probability of needing service: [tex]$24\%$[/tex] or [tex]$0.24$[/tex]
- Cost with insurance: [tex]$\$[/tex]100[tex]$
- Expected cost for doctor visits: $[/tex]0.24 \times 100 = \[tex]$24$[/tex]
3. Medication:
- Probability of needing service: [tex]$12\%$[/tex] or [tex]$0.12$[/tex]
- Cost with insurance: [tex]$\$[/tex]40[tex]$
- Expected cost for medication: $[/tex]0.12 \times 40 = \[tex]$4.8$[/tex]
Sum of all expected costs with insurance:
- Expected value with insurance: [tex]$1580 + 24 + 4.8 = \$[/tex]1608.8[tex]$
### Final Answer:
The expected value of health care without insurance is \$[/tex][tex]\(\boxed{364.8}\)[/tex]
The expected value of health care with insurance is \$[tex]\(\boxed{1608.8}\)[/tex]
