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Regina used her past health history and information about her doctor visits to create this table to compare health costs with and without insurance.

\begin{tabular}{|l|c|c|c|}
\hline Description of Service & \begin{tabular}{c}
Probability of Needing \\
the Service
\end{tabular} & \begin{tabular}{c}
Cost with Insurance \\
Plan
\end{tabular} & \begin{tabular}{c}
Cost without Insurance \\
Plan
\end{tabular} \\
\hline annual premium & [tex]$100 \%$[/tex] & [tex]$\$[/tex] 1,580[tex]$ & $[/tex]\[tex]$ 0$[/tex] \\
\hline four doctor visits & [tex]$24 \%$[/tex] & [tex]$\$[/tex] 100[tex]$ & $[/tex]\[tex]$ 1,400$[/tex] \\
\hline medication & [tex]$12 \%$[/tex] & [tex]$\$[/tex] 40[tex]$ & $[/tex]\[tex]$ 240$[/tex] \\
\hline
\end{tabular}

What is the expected value of each option?

The expected value of health care without insurance is [tex]$\$[/tex][tex]$ $[/tex]\square[tex]$

The expected value of health care with insurance is $[/tex]\[tex]$[/tex] [tex]$\square$[/tex]



Answer :

GinnyAnswer
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]

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