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Gareth is choosing a car insurance plan. Based on his driving history and the traffic where he lives, Gareth estimates that there is a 15% 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?




Plan		DeductibleCollisionComprehensivePremium Total

A		\$300\$516\$240\$756

B		\$500\$445\$210\$655

C		\$1,000\$350\$184\$534

D		\$2,500\$288\$124\$412





Aplan A
Bplan B
Cplan C
Dplan D

Verified Answer :

  • Calculate the expected cost for each plan using the formula: \(\text{Expected Cost} = \text{Premium} + 0.15 \times \text{Deductible}\).
  • Obtain the expected costs: Plan A = \(\$801\), Plan B = \(\$730\), Plan C = \(\$684\), and Plan D = \(\$787\).
  • Compare these values to identify the minimum.
  • The plan that saves Gareth the most money is \(\boxed{\text{plan C}}\).

Explanation

  1. Analyze the cost structure and expected value formula

To determine which insurance plan is the most cost-effective for Gareth, we calculate the expected value (EV) of the total cost for each plan. The expected total cost is defined as the sum of the guaranteed premium and the expected cost associated with a collision (which is the deductible multiplied by the probability of a collision occurring).
We are given:

  • Probability of collision: \(P(\text{collision}) = 0.15\)
  • Total cost for each plan: \(\text{Expected Cost} = \text{Premium Total} + (P(\text{collision}) \times \text{Deductible})\)
  1. Calculate the expected cost for each plan

Using the provided table, we calculate the expected total cost for each plan:

  1. Plan A:

\(\text{Premium Total} = \$756\), \(\text{Deductible} = \$300\)
\(\text{Expected Cost}_A = 756 + (0.15 \times 300) = 756 + 45 = \$801\)

  1. Plan B:

\(\text{Premium Total} = \$655\), \(\text{Deductible} = \$500\)
\(\text{Expected Cost}_B = 655 + (0.15 \times 500) = 655 + 75 = \$730\)

  1. Plan C:

\(\text{Premium Total} = \$534\), \(\text{Deductible} = \$1,000\)
\(\text{Expected Cost}_C = 534 + (0.15 \times 1,000) = 534 + 150 = \$684\)

  1. Plan D:

\(\text{Premium Total} = \$412\), \(\text{Deductible} = \$2,500\)
\(\text{Expected Cost}_D = 412 + (0.15 \times 2,500) = 412 + 375 = \$787\)

  1. Compare costs and select the best plan

Comparing the calculated expected costs:

  • Plan A: \(\$801\)
  • Plan B: \(\$730\)
  • Plan C: \(\$684\)
  • Plan D: \(\$787\)

The minimum expected cost is \(\$684\), which corresponds to Plan C. Therefore, Plan C is the most likely to save Gareth the most money.

Examples

Understanding expected value is essential for making smart financial decisions, like choosing insurance or evaluating investments. For example, when you buy car insurance, you're not just looking at the sticker price; you're weighing the guaranteed monthly premium against the potential out-of-pocket costs you might face if an accident occurs. By calculating the expected value, you can mathematically determine which option minimizes your long-term risk and expense, just as we did to find the best plan for Gareth.

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