Answer :
To determine which car insurance plan is most likely to save Brenda the most money based on expected value, we need to calculate the expected cost for each plan. Expected cost is determined by taking into account both the fixed premium and the potential cost of a collision, adjusted for the probability of the collision occurring.
Let's break down the costs for each plan:
1. Plan A:
- Deductible: \[tex]$300 - Collision: \$[/tex]500
- Comprehensive: \[tex]$234 - Premium Total: \$[/tex]734
Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost))
Expected Cost = \[tex]$734 + \(0.2 \times (\$[/tex]300 + \[tex]$500)\) Expected Cost = \$[/tex]734 + [tex]\(0.2 \times \$800\)[/tex]
Expected Cost = \[tex]$734 + \$[/tex]160
Expected Cost = \[tex]$894 2. Plan B: - Deductible: \$[/tex]500
- Collision: \[tex]$430 - Comprehensive: \$[/tex]212
- Premium Total: \[tex]$642 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]642 + [tex]\(0.2 \times (\$500 + \$430)\)[/tex]
Expected Cost = \[tex]$642 + \(0.2 \times \$[/tex]930\)
Expected Cost = \[tex]$642 + \$[/tex]186
Expected Cost = \[tex]$828 3. Plan C: - Deductible: \$[/tex]1,000
- Collision: \[tex]$340 - Comprehensive: \$[/tex]175
- Premium Total: \[tex]$515 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]515 + [tex]\(0.2 \times (\$1,000 + \$340)\)[/tex]
Expected Cost = \[tex]$515 + \(0.2 \times \$[/tex]1,340\)
Expected Cost = \[tex]$515 + \$[/tex]268
Expected Cost = \[tex]$783 4. Plan D: - Deductible: \$[/tex]2,500
- Collision: \[tex]$278 - Comprehensive: \$[/tex]127
- Premium Total: \[tex]$405 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]405 + [tex]\(0.2 \times (\$2,500 + \$278)\)[/tex]
Expected Cost = \[tex]$405 + \(0.2 \times \$[/tex]2,778\)
Expected Cost = \[tex]$405 + \$[/tex]555.60
Expected Cost = \[tex]$960.60 Now, we compare the expected costs: - Plan A: \$[/tex]894
- Plan B: \[tex]$828 - Plan C: \$[/tex]783
- Plan D: \[tex]$960.60 The lowest expected cost is \$[/tex]783 for Plan C.
Therefore, the plan that is most likely to save Brenda the most money, based on expected value, is:
C. plan C
Let's break down the costs for each plan:
1. Plan A:
- Deductible: \[tex]$300 - Collision: \$[/tex]500
- Comprehensive: \[tex]$234 - Premium Total: \$[/tex]734
Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost))
Expected Cost = \[tex]$734 + \(0.2 \times (\$[/tex]300 + \[tex]$500)\) Expected Cost = \$[/tex]734 + [tex]\(0.2 \times \$800\)[/tex]
Expected Cost = \[tex]$734 + \$[/tex]160
Expected Cost = \[tex]$894 2. Plan B: - Deductible: \$[/tex]500
- Collision: \[tex]$430 - Comprehensive: \$[/tex]212
- Premium Total: \[tex]$642 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]642 + [tex]\(0.2 \times (\$500 + \$430)\)[/tex]
Expected Cost = \[tex]$642 + \(0.2 \times \$[/tex]930\)
Expected Cost = \[tex]$642 + \$[/tex]186
Expected Cost = \[tex]$828 3. Plan C: - Deductible: \$[/tex]1,000
- Collision: \[tex]$340 - Comprehensive: \$[/tex]175
- Premium Total: \[tex]$515 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]515 + [tex]\(0.2 \times (\$1,000 + \$340)\)[/tex]
Expected Cost = \[tex]$515 + \(0.2 \times \$[/tex]1,340\)
Expected Cost = \[tex]$515 + \$[/tex]268
Expected Cost = \[tex]$783 4. Plan D: - Deductible: \$[/tex]2,500
- Collision: \[tex]$278 - Comprehensive: \$[/tex]127
- Premium Total: \[tex]$405 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]405 + [tex]\(0.2 \times (\$2,500 + \$278)\)[/tex]
Expected Cost = \[tex]$405 + \(0.2 \times \$[/tex]2,778\)
Expected Cost = \[tex]$405 + \$[/tex]555.60
Expected Cost = \[tex]$960.60 Now, we compare the expected costs: - Plan A: \$[/tex]894
- Plan B: \[tex]$828 - Plan C: \$[/tex]783
- Plan D: \[tex]$960.60 The lowest expected cost is \$[/tex]783 for Plan C.
Therefore, the plan that is most likely to save Brenda the most money, based on expected value, is:
C. plan C