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Select the correct answer.

Cameron is choosing a car insurance plan. Based on his driving history and traffic where he lives, Cameron estimates that there is a [tex]$25 \%$[/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 plan detailed in the table is most likely to save Cameron the most money, based on expected value?

\begin{tabular}{|l|c|c|c|c|}
\hline Plan & Deductible & Collision & Comprehensive & Premium Total \\
\hline A & [tex]$\$[/tex] 300[tex]$ & $[/tex]\[tex]$ 525$[/tex] & [tex]$\$[/tex] 239[tex]$ & $[/tex]\[tex]$ 764$[/tex] \\
\hline B & [tex]$\$[/tex] 500[tex]$ & $[/tex]\[tex]$ 460$[/tex] & [tex]$\$[/tex] 215[tex]$ & $[/tex]\[tex]$ 675$[/tex] \\
\hline C & [tex]$\$[/tex] 1,000[tex]$ & $[/tex]\[tex]$ 375$[/tex] & [tex]$\$[/tex] 185[tex]$ & $[/tex]\[tex]$ 560$[/tex] \\
\hline D & [tex]$\$[/tex] 2,500[tex]$ & $[/tex]\[tex]$ 300$[/tex] & [tex]$\$[/tex] 136[tex]$ & $[/tex]\[tex]$ 436$[/tex] \\
\hline
\end{tabular}

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



Answer :

GinnyAnswer
Let's analyze the car insurance plans to find the one that is most likely to save Cameron the most money, based on expected value.

First, we'll identify the key components for each plan:
- Deductible: The amount Cameron needs to pay out of pocket in the event of a collision.
- Collision Coverage: The amount the insurance covers after the deductible.
- Premium Total: The total annual payment to the insurance company.

Cameron estimates a 25% chance of having a car collision this year, which means we need to calculate the expected total cost for each plan, factoring in the deductible and the premium total.

For each plan, the expected cost can be calculated as:

[tex]\[ \text{Expected Cost} = \text{Premium Total} + (\text{Probability of Collision} \times (\text{Deductible} + \text{Collision Coverage})) \][/tex]

Let's break down the calculations for each plan:

1. Plan A:
- Deductible: \[tex]$300 - Premium Total: \$[/tex]764
- Collision Coverage: \[tex]$525 - Probability of Collision: 25% or 0.25 \[ \text{Expected Cost}_A = 764 + 0.25 \times (300 + 525) \] \[ = 764 + 0.25 \times 825 \] \[ = 764 + 206.25 \] \[ = 970.25 \] 2. Plan B: - Deductible: \$[/tex]500
- Premium Total: \[tex]$675 - Collision Coverage: \$[/tex]460
- Probability of Collision: 25% or 0.25

[tex]\[ \text{Expected Cost}_B = 675 + 0.25 \times (500 + 460) \][/tex]
[tex]\[ = 675 + 0.25 \times 960 \][/tex]
[tex]\[ = 675 + 240 \][/tex]
[tex]\[ = 915 \][/tex]

3. Plan C:
- Deductible: \[tex]$1,000 - Premium Total: \$[/tex]560
- Collision Coverage: \[tex]$375 - Probability of Collision: 25% or 0.25 \[ \text{Expected Cost}_C = 560 + 0.25 \times (1000 + 375) \] \[ = 560 + 0.25 \times 1375 \] \[ = 560 + 343.75 \] \[ = 903.75 \] 4. Plan D: - Deductible: \$[/tex]2,500
- Premium Total: \[tex]$436 - Collision Coverage: \$[/tex]300
- Probability of Collision: 25% or 0.25

[tex]\[ \text{Expected Cost}_D = 436 + 0.25 \times (2500 + 300) \][/tex]
[tex]\[ = 436 + 0.25 \times 2800 \][/tex]
[tex]\[ = 436 + 700 \][/tex]
[tex]\[ = 1136 \][/tex]

Now that we have calculated the expected costs for each plan, we can compare them:

- Plan A: \[tex]$970.25 - Plan B: \$[/tex]915
- Plan C: \[tex]$903.75 - Plan D: \$[/tex]1136

The plan with the lowest expected cost is Plan C with an expected cost of \$903.75.

Therefore, the correct answer is:
C. Plan C