Let’s solve the problem step by step. ## **Given:** - **Cost per repair:** $237 - **Service option 1:** $507 covers all repairs - **Service option 2:** $381 covers any repairs after the first one - **Breakdown probabilities:** - 0 breakdowns: 0.10 - 1 breakdown: 0.20 - 2 breakdowns: 0.35 - 3 breakdowns: 0.35 --- ## **Step 1: Calculate expected cost for each option** ### **A. Pay for all repairs** Expected cost = Σ (Probability × Cost for that number of breakdowns) \[ \text{Expected Cost} = (0.10 \times 0 \times 237) + (0.20 \times 1 \times 237) + (0.35 \times 2 \times 237) + (0.35 \times 3 \times 237) \] \[ = 0 + 0.20 \times 237 + 0.35 \times 474 + 0.35 \times 711 \] \[ = 0 + 47.4 + 165.9 + 248.85 \] \[ = 462.15 \] **Rounded to the nearest whole number:** **462** --- ### **B. Service Option 1 ($507 covers all repairs)** - No matter how many breakdowns, you pay $507. - Expected cost = $507 --- ### **C. Service Option 2 ($381 + first repair at $237)** - For 0 breakdowns: $0 - For 1 breakdown: $237 - For 2 or more breakdowns: $237 (first repair) + $381 = $618 \[ \text{Expected Cost} = (0.10 \times 0) + (0.20 \times 237) + (0.35 \times 618) + (0.35 \times 618) \] \[ = 0 + 47.4 + 216.3 + 216.3 \] \[ = 479.9 \] **Rounded to the nearest whole number:** **480** --- ## **Step 2: Summary Table** | Option | Expected Cost | |-------------------------------|---------------| | Pay for all repairs | **462** | | Service option number 1 | **507** | | Service option number 2 | **480** | --- ## **Step 3: Final Answer** ### **The lowest expected cost is to "Pay for all repairs" (Expected cost = $462).** **So, the correct answer is:** > **○ Pay for all repairs**

Let's analyze the problem carefully and go through each step with detailed explanations. --- ### **Understanding the problem:** - The machine can break down 0, 1, 2, or 3 times, with given probabilities. - The costs associated with repairs are: - **Cost per repair:** $237 - **Service options:** - **Option 1:** Pay a flat fee of **$507** which covers all repairs regardless of the number of breakdowns. - **Option 2:** Pay **$381** plus **$237** for the first repair, and for additional repairs, pay **$381** each time (i.e., after the first repair, each additional repair costs $381). The goal is to determine which option results in the lowest *expected* cost based on the probabilities of breakdowns. --- ### **Step 1: Understand the breakdown probabilities** | Number of breakdowns | Probability | |------------------------|--------------| | 0 | 0.10 | | 1 | 0.20 | | 2 | 0.35 | | 3 | 0.35 | --- ### **Step 2: Calculate the expected cost for each payment option** --- ### **Option A: Pay for all repairs** - Cost per repair = $237 - For each number of breakdowns, total repair cost = number of breakdowns × $237 **Expected Cost Calculation:** \[ \text{Expected Cost} = \sum (\text{Probability} \times \text{Cost for that number of breakdowns}) \] Specifically: \[ = (0.10 \times 0 \times 237) + (0.20 \times 1 \times 237) + (0.35 \times 2 \times 237) + (0.35 \times 3 \times 237) \] Let's compute each term: - For 0 breakdowns: \(0.10 \times 0 \times 237 = 0\) - For 1 breakdown: \(0.20 \times 1 \times 237 = 0.20 \times 237 = 47.4\) - For 2 breakdowns: \(0.35 \times 2 \times 237 = 0.35 \times 474 = 165.9\) - For 3 breakdowns: \(0.35 \times 3 \times 237 = 0.35 \times 711 = 248.85\) Adding these: \[ 0 + 47.4 + 165.9 + 248.85 = 462.15 \] **Expected cost for Option A:** **$462** (rounded to nearest dollar). --- ### **Option B: Service option number 1 ($507 covers all repairs)** - Regardless of the number of breakdowns, the cost is **$507**. - **Expected cost:** **$507** --- ### **Option C: Service option number 2** - Cost structure: - First repair: **$237** - After that, each additional repair costs **$381** **Breakdown of costs:** | Number of breakdowns | Cost for repairs | |------------------------|------------------| | 0 | $0 | | 1 | $237 | | 2 | $237 + $381 = $618 | | 3 | $237 + 2 × $381 = $237 + $762 = $999 | **Expected Cost Calculation:** \[ = (0.10 \times 0) + (0.20 \times 237) + (0.35 \times 618) + (0.35 \times 999) \] Calculations: - For 0 breakdowns: \(0.10 \times 0 = 0\) - For 1 breakdown: \(0.20 \times 237 = 47.4\) - For 2 breakdowns: \(0.35 \times 618 = 216.3\) - For 3 breakdowns: \(0.35 \times 999 = 349.65\) Adding these: \[ 0 + 47.4 + 216.3 + 349.65 = 613.35 \] **Expected cost for Option C:** **$613** (rounded to nearest dollar). --- ### **Step 3: Compare the expected costs** | Option | Expected Cost | |----------------------------------|----------------| | Pay for all repairs | **$462** | | Service option 1 ($507 flat fee) | **$507** | | Service option 2 ($381 + costs) | **$613** | --- ### **Final conclusion:** The lowest expected cost is **$462** when choosing to **pay for all repairs**. --- ### **Final Answer:** **The best choice is to "Pay for all repairs."** --- **Summary:** - Calculated the expected costs for each option based on probabilities and costs. - Found that paying for all repairs results in the lowest expected cost ($462). - Therefore, the optimal choice is to **pay for all repairs**. --- **If you have any further questions or need clarification, feel free to ask!**