Answered

Select the correct answer.

Gareth is choosing a car insurance plan. Based on his driving history and the traffic where he lives, Gareth estimates that there is a [tex]$15\%$[/tex] chance he will have a car collision this year. In each plan, the insurance will cover the full cost of the collision after the deductible is paid.

Which of the plans detailed in the table is most likely to save Gareth the most money, based on expected value?

\begin{tabular}{|l|c|c|c|c|}
\hline
Plan & Deductible & Collision & Comprehensive & Premium Total \\
\hline
A & \[tex]$300 & \$[/tex]516 & \[tex]$240 & \$[/tex]756 \\
\hline
B & \[tex]$500 & \$[/tex]445 & \[tex]$210 & \$[/tex]655 \\
\hline
C & \[tex]$1,000 & \$[/tex]350 & \[tex]$184 & \$[/tex]534 \\
\hline
D & \[tex]$2,500 & \$[/tex]288 & \[tex]$124 & \$[/tex]412 \\
\hline
\end{tabular}

A. Plan A
B. Plan B
C. Plan C
D. Plan D



Answer :

GinnyAnswer
To determine which car insurance plan is most likely to save Gareth the most money based on expected value, we need to consider both the deductible of each plan and the total premium cost. We aim to calculate the total expected cost for each plan and select the one with the lowest expected cost.

1. Plan A
- Deductible: [tex]$300 - Premium Total: $[/tex]756
- Probability of Collision: 15%

Expected collision cost:
[tex]\[ 0.15 \times 300 = 45 \][/tex]

Total expected cost for Plan A:
[tex]\[ 45 + 756 = 801 \][/tex]

2. Plan B
- Deductible: [tex]$500 - Premium Total: $[/tex]655
- Probability of Collision: 15%

Expected collision cost:
[tex]\[ 0.15 \times 500 = 75 \][/tex]

Total expected cost for Plan B:
[tex]\[ 75 + 655 = 730 \][/tex]

3. Plan C
- Deductible: [tex]$1,000 - Premium Total: $[/tex]534
- Probability of Collision: 15%

Expected collision cost:
[tex]\[ 0.15 \times 1000 = 150 \][/tex]

Total expected cost for Plan C:
[tex]\[ 150 + 534 = 684 \][/tex]

4. Plan D
- Deductible: [tex]$2,500 - Premium Total: $[/tex]412
- Probability of Collision: 15%

Expected collision cost:
[tex]\[ 0.15 \times 2500 = 375 \][/tex]

Total expected cost for Plan D:
[tex]\[ 375 + 412 = 787 \][/tex]

Comparing the total expected costs:

- Plan A: [tex]$801 - Plan B: $[/tex]730
- Plan C: [tex]$684 - Plan D: $[/tex]787

The plan with the lowest expected cost is Plan C with a total expected cost of $684.

Therefore, the correct answer is:
C. plan C