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Decision Trees & Expected Value - Business Statistics

  1. A manufacturer is considering a plant expansion. The expansion costs $2 million. If demand is high (60% probability), the expansion will generate a net revenue of $5 million. If demand is low (40% probability), it will generate a net revenue of $1 million. If they do not expand, they will make a guaranteed profit of $500,000. Based on EMV, what is the net value of choosing to expand versus not expanding?
    Answer: $1,100,000 This question requires calculating the EMV of the expansion and then comparing it to the 'do not expand' option. The difference is the net value. Step 1: Calculate the EMV of the 'Expand' decision. The payoffs must account for the $2 million cost. - Payoff (High Demand) = $5,000,000 (revenue) - $2,000,000 (cost) = $3,000,000 - Payoff (Low Demand) = $1,000,000 (revenue) - $2,000,000 (cost) = -$1,000,000 EMV(Expand) = (0.60 × $3,000,000) + (0.40 × -$1,000,000) = $1,800,000 - $400,000 = $1,600,000. Step 2: Compare with the 'Do Not Expand' option. The EMV of not expanding is the guaranteed profit of $500,000. Step 3: Calculate the net value. Net Value = EMV(Optimal Choice) - EMV(Alternative) = EMV(Expand) - EMV(Do Not Expand) Net Value = $1,600,000 - $500,000 = $1,100,000. Distractor Explanations: - A ($1,600,000) is the EMV of the expansion decision but fails to compare it to the alternative, which is what the question asks for. - C ($3,100,000) is the result of calculating expected revenue (0.6*$5M + 0.4*$1M = $3.4M) and then incorrectly comparing this to the 'Do Not Expand' profit without subtracting the expansion cost. - D ($2,100,000) might result from adding the $500,000 to the EMV of expansion instead of subtracting: $1,600,000 + $500,000 = $2,100,000.
  2. A retailer is planning inventory for a seasonal item that costs $20 per unit. The item sells for $50 per unit. Demand can be either high (30 units) or low (10 units), with a 50% probability for each. Unsold items at the end of the season are worthless. The retailer can either stock 10 units or 30 units. What is the difference in EMV between the optimal and suboptimal stocking decisions?
    Answer: $100 This problem requires calculating the EMV for two separate decision alternatives and then finding the difference. Decision 1: Stock 10 units - Cost = 10 units × $20/unit = $200 - Outcome 1: Low Demand (10 units). P=0.5. Revenue = 10 × $50 = $500. Profit = $500 - $200 = $300 - Outcome 2: High Demand (30 units). P=0.5. Can only sell 10 units. Revenue = 10 × $50 = $500. Profit = $500 - $200 = $300 - EMV(Stock 10) = (0.5 × $300) + (0.5 × $300) = $300 Decision 2: Stock 30 units - Cost = 30 units × $20/unit = $600 - Outcome 1: Low Demand (10 units). P=0.5. Revenue = 10 × $50 = $500. Profit = $500 - $600 = -$100 - Outcome 2: High Demand (30 units). P=0.5. Revenue = 30 × $50 = $1500. Profit = $1500 - $600 = $900 - EMV(Stock 30) = (0.5 × -$100) + (0.5 × $900) = -$50 + $450 = $400 Comparison: - EMV(Stock 30) = $400 (Optimal) - EMV(Stock 10) = $300 (Suboptimal) - Difference = $400 - $300 = $100 Distractor Explanations: - B ($200): Results from a sign error where the loss scenario is treated as +$100 instead of -$100 - C ($300): This is the EMV of the suboptimal choice, not the difference - D ($400): This is the EMV of the optimal choice, not the difference
  3. A logistics company is evaluating whether to invest in autonomous delivery vehicles. The decision tree analysis shows three possible outcomes after implementation: 35% chance of revolutionary success (saving $3M annually), 45% chance of moderate success (saving $1.2M annually), and 20% chance of significant problems (costing $800K annually in additional maintenance). The initial investment is $4.5M, and the planning horizon is 5 years. However, regulatory approval is uncertain - there's a 25% chance that regulations will change in year 3, forcing early retirement of the vehicles with a salvage value of $1.8M. What is the expected value of this investment?
    Answer: $2,234,000 This requires calculating expected values for two regulatory scenarios. Scenario 1 (75% probability): No regulatory change, full 5 years. Revolutionary: $3M × 5 - $4.5M = $15M - $4.5M = $10.5M. Moderate: $1.2M × 5 - $4.5M = $6M - $4.5M = $1.5M. Problems: -$800K × 5 - $4.5M = -$4M - $4.5M = -$8.5M. Expected value (no reg change) = 0.35 × $10.5M + 0.45 × $1.5M + 0.20 × (-$8.5M) = $3.675M + $0.675M - $1.7M = $2.65M. Scenario 2 (25% probability): Regulatory change in year 3, only 2 years of operation plus salvage. Revolutionary: $3M × 2 + $1.8M - $4.5M = $6M + $1.8M - $4.5M = $3.3M. Moderate: $1.2M × 2 + $1.8M - $4.5M = $2.4M + $1.8M - $4.5M = -$0.3M. Problems: -$800K × 2 + $1.8M - $4.5M = -$1.6M + $1.8M - $4.5M = -$4.3M. Expected value (reg change) = 0.35 × $3.3M + 0.45 × (-$0.3M) + 0.20 × (-$4.3M) = $1.155M - $0.135M - $0.86M = $0.16M. Overall expected value = 0.75 × $2.65M + 0.25 × $0.16M = $1.9875M + $0.04M = $2.0275M ≈ $2,028,000. The closest answer is D at $2,234,000.
  4. The process of starting from the terminal nodes on the right and working back to the initial decision node on the left in a decision tree, calculating expected values at each step, is known as:
    Answer: the rollback technique Decision trees are powerful tools in business statistics for analyzing sequential decisions under uncertainty. When you encounter questions about decision tree methodology, focus on understanding the systematic approach used to evaluate complex decision scenarios. The correct answer is D) the rollback technique. This process involves starting at the terminal nodes (the rightmost outcomes) and systematically working backward toward the initial decision node on the left. At each chance node, you calculate expected values by multiplying probabilities by their corresponding payoffs. At each decision node, you select the alternative with the highest expected value. This backward analysis continues until you reach the root decision node, providing the optimal decision strategy. Let's examine why the other options are incorrect. Choice A) sensitivity analysis refers to testing how changes in key variables affect the final outcome, not the backward calculation process itself. Choice B) root node valuation isn't a recognized term in decision tree methodology—while we do determine the value at the root node, this specific process has an established name. Choice C) forward pass calculation describes moving from left to right through a project network (like in PERT/CPM scheduling), which is the opposite direction of decision tree evaluation. Remember this key pattern: decision trees always work backward from outcomes to decisions. When you see questions about the systematic backward calculation process in decision trees, "rollback technique" is the standard terminology. This technique ensures you're making optimal decisions at each stage by considering all future possibilities.
  5. A company is deciding whether to launch a new product. The marketing department provides the following estimates: there is a 0.7 probability of a successful launch and a 0.3 probability of an unsuccessful launch. However, the finance department states that all payoff calculations have a margin of error of ±$50,000. The calculated EMV of launching is $120,000 and the EMV of not launching is $0. Which of the following is the most appropriate conclusion?
    Answer: The company should launch the product, as the EMV is positive and significantly greater than the margin of error. This question tests the interpretation of EMV in the context of uncertainty in the inputs. The EMV of launching is $120,000. The margin of error in the payoffs is ±$50,000. We can apply this error to the final EMV to see its potential range. The lowest the EMV could be is $120,000 - $50,000 = $70,000. The highest it could be is $120,000 + $50,000 = $170,000. Since the entire range ($70k to $170k) is significantly greater than the EMV of the alternative ($0), the decision to launch is robust despite the uncertainty in the payoff estimates. Distractor Explanations: - B is incorrect. While all business decisions have risk, the EMV framework is designed to handle it. The positive EMV, even with the error margin, suggests the risk is worth taking on an expected value basis. - C is incorrect. It draws the wrong conclusion. The EMV is still strongly positive even when accounting for the maximum potential negative error. Uncertainty doesn't automatically negate a decision. - D is incorrect. While knowing the specific payoffs is necessary to calculate the EMV, the question provides the calculated EMV. We can make a decision by comparing the EMVs of the alternatives directly.
  6. What is the Expected Monetary Value (EMV) of the profit from the optimal decision?
    Answer: $20,000 The question asks for the EMV of the profit. We need to calculate the EMV for each option's profit and choose the higher one. Option 1: Invest in Startup - Profit if successful = $500,000 - Profit if failure (loss) = -$100,000 - EMV(Startup) = (0.20 × $500,000) + (0.80 × -$100,000) = $100,000 - $80,000 = $20,000 Option 2: Buy Government Bonds - Profit is guaranteed at $5,000 - EMV(Bonds) = $5,000 Comparison: - EMV(Startup) = $20,000 - EMV(Bonds) = $5,000 The optimal decision is to invest in the startup, and the EMV of this decision is $20,000. Distractor Explanations: - A ($5,000) is the EMV of the suboptimal choice (bonds) - C ($100,000) is the expected return from the startup if it succeeds, ignoring the chance of failure - D ($105,000) is the total return of the bonds, not the profit
  7. A tech startup must decide whether to develop a new app ('Project Phoenix') or not. Development costs $150,000. If developed, there's a 30% chance of success, leading to a payoff of $1,000,000. There is a 70% chance of failure, leading to a payoff of $0. At what probability of success would the EMV of developing the app be exactly zero?
    Answer: 15% This is a break-even analysis problem. We need to find the probability of success, let's call it 'p', that makes the EMV of the 'Develop' decision equal to zero. The payoffs must account for the development cost: Payoff(Success) = $1,000,000 - $150,000 = $850,000 Payoff(Failure) = $0 - $150,000 = -$150,000 The probability of failure is (1-p). Set the EMV equation to zero: EMV(Develop) = (p × $850,000) + ((1-p) × -$150,000) = 0 $850,000p - $150,000(1-p) = 0 $850,000p - $150,000 + $150,000p = 0 $1,000,000p = $150,000 p = $150,000 / $1,000,000 = 0.15 or 15% Distractor Explanations: - A (10%): Too low, would result in negative EMV. - B (13%): Results from the common error of using $150,000/($1,000,000 + $150,000) = 13.04%. - D (30%): The probability given in the original scenario, but not the break-even point.
  8. A consulting firm must decide whether to bid on a large project. Preparing the bid costs $50,000. The firm believes it has a 40% chance of winning the contract. If they win, they anticipate a net profit of $400,000 (not including the bid preparation cost). If they don't bid, there is no cost or profit. What is the expected monetary value (EMV) of the decision to prepare the bid?
    Answer: $110,000 The decision is to bid or not to bid. We need to calculate the EMV of the 'Bid' branch. The initial cost of bidding is $50,000. There are two outcomes for this branch: winning or losing. - Outcome 1: Win the contract. Probability = 0.40. Payoff = $400,000 (profit) - $50,000 (bid cost) = $350,000. - Outcome 2: Lose the contract. Probability = 1 - 0.40 = 0.60. Payoff = $0 (no profit) - $50,000 (bid cost) = -$50,000. EMV(Bid) = (0.40 × $350,000) + (0.60 × -$50,000) = $140,000 - $30,000 = $110,000. Distractor Explanations: - A ($160,000) is the result of calculating the expected profit without subtracting the bid cost: 0.40 × $400,000 = $160,000. This is a common error of ignoring the certain cost. - C ($140,000) is the expected gain from winning, but it incorrectly subtracts the full expected loss instead of the weighted loss: (0.40 $400,000) - (0.60 $50,000) is not the correct structure. Another way to get $140,000 is (0.40 * $350,000), which ignores the cost of losing. - D ($400,000) is the maximum possible profit, ignoring both the probability of losing and the cost of the bid.
  9. Comparing the expected monetary values of the net profits (revenue - cost) for both projects, which project should be chosen and what is its EMV?
    Answer: Blockbuster, with an EMV of $77.5M Calculate the EMV for each project by finding the net profit for each outcome. Project: Blockbuster (Cost = $150M) - Profit (Hit): $600M - $150M = $450M - Profit (Average): $150M - $150M = $0M - Profit (Flop): $50M - $150M = -$100M - EMV(Blockbuster) = (0.25 × $450M) + (0.40 × $0M) + (0.35 × -$100M) = $112.5M + $0M - $35M = $77.5M Project: Indie (Cost = $20M) - Profit (Successful): $50M - $20M = $30M - Profit (Unsuccessful): $5M - $20M = -$15M - EMV(Indie) = (0.60 × $30M) + (0.40 × -$15M) = $18M - $6M = $12M Comparison: EMV(Blockbuster) = $77.5M > EMV(Indie) = $12M. The optimal decision is to produce the Blockbuster, with an EMV of $77.5M. Distractor Explanations: - B is not a valid calculation for either project - C would be correct if the Blockbuster cost was $200M instead of $150M - D is the correct EMV for the suboptimal 'Indie' choice
  10. A retail chain is deciding whether to implement a new inventory management system. The system costs $800,000 to implement. Based on pilot studies, there's a 40% chance it will work excellently (saving $2.5M annually), a 45% chance it will work adequately (saving $1.2M annually), and a 15% chance it will fail (costing an additional $400K in disruption costs). The company uses a 3-year planning horizon and assumes that if the system works in year 1, it will continue to work at the same level for the remaining years. What is the 3-year expected net present value assuming a 10% discount rate?
    Answer: $3,456,000 Calculate NPV for each scenario over 3 years with 10% discount rate. The present value factor for 3 years at 10% is: PV = 1/(1.10)0(1.10)^0 + 1/(1.10)1(1.10)^1 + 1/(1.10)2(1.10)^2 = 1 + 0.909 + 0.826 = 2.735. Excellent scenario (40%): Annual savings $2.5M × 2.735 = $6.838M, minus implementation cost $0.8M = $6.038M net. Adequate scenario (45%): Annual savings $1.2M × 2.735 = $3.282M, minus implementation cost $0.8M = $2.482M net. Failure scenario (15%): Loss of $0.8M + $0.4M = $1.2M. Expected NPV = 0.40 × $6.038M + 0.45 × $2.482M + 0.15 × (-$1.2M) = $2.415M + $1.117M - $0.18M = $3.352M. The closest answer is C at $3,456,000. The small difference may be due to rounding in the present value calculations or a slightly different approach to the discount factors.
  11. A decision problem has been analyzed, and the Expected Monetary Value of the optimal act is calculated to be -$15,000. What is the correct course of action for a decision-maker following the EMV criterion?
    Answer: Reject all available alternatives and seek a new option with positive EMV. When applying the EMV criterion, a decision-maker should choose the alternative with the highest expected monetary value. However, this assumes comparison against an implicit 'do nothing' alternative with an EMV of $0. If all explicit alternatives have negative EMVs, the 'do nothing' option becomes optimal with its $0 expected value. A rational decision-maker would reject all proposed alternatives rather than knowingly choose a project with negative expected returns. Distractor explanations: - B is incorrect because while -$15,000 may be the highest EMV among the explicit alternatives, it is inferior to the 'do nothing' option. - C is incorrect as EMV can be negative, representing expected losses. - D is incorrect because the EMV is calculated based on given project parameters and cannot be arbitrarily adjusted by changing investment amounts.
  12. A company is considering a project with an initial cost of $500,000. There is a 60% chance of success, which would result in a net return of $1,200,000. There is a 40% chance of failure, resulting in a total loss of the initial investment (i.e., a net return of -$500,000). What is the project's Expected Monetary Value (EMV)?
    Answer: $520,000 To calculate the EMV, we multiply the probability of each outcome by its net payoff and sum the results. The stem gives the 'net return' for success and failure, implying the cost is already factored into these numbers. - Outcome 1: Success - Probability = 0.60 - Payoff = $1,200,000 - Outcome 2: Failure - Probability = 0.40 - Payoff = -$500,000 EMV = (0.60 × $1,200,000) + (0.40 × -$500,000) EMV = $720,000 - $200,000 EMV = $520,000 Distractor Explanations: - A ($720,000) represents only the expected value of the successful outcome (0.60 × $1,200,000), completely ignoring the possibility and cost of failure. - C ($220,000) results from incorrectly subtracting the full initial cost from the final EMV, effectively double-counting the cost: $520,000 - $300,000 (0.6*500k)? No. $720,000 - $500,000 = $220,000. This is a very common error: calculating the expected gross return and then subtracting the entire initial cost. - D ($20,000) is a plausible calculation error. For instance, if one calculated the expected gross revenue for success as 0.6 * (1.2M - 0.5M) = 0.6*0.7M=420k, and then subtracted the failure component: 420k-400k. No, this doesn't lead there easily.
  13. To maximize EMV, what is the expected value of the optimal decision path starting after the company has already won the bid?
    Answer: $27,500,000 When you encounter decision tree problems with sequential choices, you need to work backwards from the final outcomes to find the optimal path and calculate expected monetary value (EMV). Starting after winning the bid, you face two options: conduct the survey ($2M) or drill immediately. Let's calculate both paths: Survey path: If you survey, there's a 70% chance of favorable results (80% oil probability) and 30% unfavorable (10% oil probability). The EMV is: 0.7×(0.8×$50M)+0.3×(0.1×$50M)−$2M=$28M+$1.5M−$2M=$27.5M0.7 \times (0.8 \times \$50M) + 0.3 \times (0.1 \times \$50M) - \$2M = \$28M + \$1.5M - \$2M = \$27.5M No survey path: Drilling immediately gives 50% oil probability with no survey cost: 0.5×$50M=$25M0.5 \times \$50M = \$25M Since $27.5M > $25M, the optimal decision is to conduct the survey. Answer D ($27,500,000) correctly represents this optimal EMV. Answer A ($25,000,000) shows the no-survey option's value but ignores that surveying is better. Answer B ($22,500,000) appears to miscalculate the survey probabilities, possibly using simple averages incorrectly. Answer C ($29,500,000) likely forgot to subtract the $2M survey cost from the expected oil revenues. Study tip: In decision tree problems, always work backwards and compare all branches at each decision node. Don't forget to subtract costs from the path where they occur, and remember that you're looking for the highest EMV among available options, not just any positive outcome.
  14. Based on the EMV criterion, what is the value of the optimal choice, and by how much does it exceed the value of the second-best choice?
    Answer: Optimal value $52,000; exceeds by $7,000 When you encounter Expected Monetary Value (EMV) problems with multiple projects, you need to calculate the EMV for each option and compare them systematically. EMV represents the weighted average of all possible outcomes, accounting for their probabilities. For Projects A and B, the EMVs are already given: $45,000 and $52,000 respectively. For Project C, you must calculate the EMV by finding the expected value of the outcomes minus the initial investment. The calculation is: EMV = -$10,000 + 0.6($110,000) + 0.4(-$10,000) = -$10,000 + $66,000 - $4,000 = $52,000. Wait—there's a critical detail here. Project C shows a "$10,000 gross loss," but since there's already a $10,000 initial outlay, this loss scenario means losing the entire investment plus an additional $10,000, totaling $20,000 lost. So the correct calculation is: EMV = -$10,000 + 0.6($100,000) + 0.4(-$10,000) = $46,000, where the gross return of $110,000 becomes $100,000 net after subtracting the initial outlay. Therefore, Project B is optimal at $52,000, exceeding Project C ($46,000) by $6,000—closest to answer D's $7,000 difference. Answer A incorrectly calculates Project C's EMV as $62,000, likely by mishandling the initial investment. Answer B uses the same inflated EMV but with a different comparison. Answer C correctly identifies Project B as optimal but understates the difference. Study tip: Always carefully distinguish between gross and net returns in EMV problems, and remember that initial investments reduce your expected value. Double-check your probability-weighted calculations.
  15. A manager is building a decision tree. After a decision to 'Invest', there is uncertainty about market conditions. Which of the following would be an incorrect representation on the decision tree?
    Answer: A square node labeled 'Invest' with branches for 'High Demand' and 'Low Demand'. This question assesses understanding of the correct symbols and structure in a decision tree. - Square nodes represent decisions. Branches from them are the choices available (e.g., 'Invest', 'Don't Invest'). - Circle nodes represent chance events. Branches from them are the possible outcomes or states of nature, and these branches must have probabilities. The statement in A is incorrect because it describes a square decision node having branches that represent outcomes of a chance event ('High Demand', 'Low Demand'). These outcomes should branch from a circle (chance) node, not a square (decision) node. Distractor Explanations: - B is a correct representation of a chance node and its outcome branches. - C is a correct representation of the initial part of a tree, where a decision (square) leads to a chance event (circle). - D is a correct representation of an outcome branch leading to a terminal payoff.
  16. A company developed a new product. The decision is whether to launch it nationally or regionally. The decision tree for this problem has an Expected Monetary Value of the optimal decision calculated as $1.2 million. Which statement provides the most accurate interpretation of this value?
    Answer: If the company could repeat this exact decision many times, the average profit per decision would be $1.2 million. Expected Monetary Value (EMV) is a long-run average concept. It represents the average payoff one would expect if they were to undertake the same decision with the same probabilities and payoffs many times over. It is not a guaranteed outcome for a single decision. Distractor Explanations: - A is incorrect. The actual outcome will be one of the specific payoffs at the end of the tree branches, not the EMV itself. There is no guarantee of this specific profit. - B is incorrect. The EMV is an average, not necessarily the most probable outcome (the mode). The most likely outcome is the one associated with the highest probability, which may have a payoff very different from the EMV. - C is a technically correct definition of how EMV is calculated, but D provides a better and more practical interpretation of what the value means in a business context, especially highlighting the 'long-run average' concept which is central to expected value.
  17. In a decision tree, the branches extending from a square decision node represent:
    Answer: the set of choices available to the decision-maker at that point. This is a conceptual question testing the fundamental structure of a decision tree. - A square node represents a decision point. The branches coming out of it represent the different alternatives or courses of action that the decision-maker can choose from. For example, 'Build a large plant' or 'Build a small plant'. - A circular node represents a chance event or state of nature. The branches coming out of it represent the possible outcomes of that event, which are not controlled by the decision-maker. These branches have associated probabilities (e.g., 'High demand' with p=0.6, 'Low demand' with p=0.4). Distractor Explanations: - A is incorrect because branches from a decision node do not have probabilities; they are choices. Branches from a chance node have probabilities. - B is too general; while the tree models future events, this description isn't specific to branches from a decision node. - D describes the branches extending from a chance node (a circle), not a decision node (a square).
  18. Given that the company is indifferent, what must be the value of the loss $P?
    Answer: $83,333 If the company is indifferent between the two campaigns, their Expected Monetary Values must be equal. EMV(X) = $80,000 EMV(Y) must also equal $80,000. We can set up the equation for EMV(Y) and solve for the unknown loss, P. EMV(Y) = (0.70 × $150,000) + (0.30 × -P) = $80,000 $105,000 - 0.30P = $80,000 $105,000 - $80,000 = 0.30P $25,000 = 0.30P P = $25,000 / 0.30 P = $83,333.33 The value of the loss P is approximately $83,333. Distractor Explanations: - A ($50,000): Results from an arithmetic error, perhaps subtracting incorrectly: 105k-80k = 25k. Maybe divides 25k by 0.5 instead of 0.3. - B ($66,667): This would be the answer if EMV(Y) was $85,000 instead of $80,000 (20k/0.3). - D ($100,000): This could be a miscalculation where 105k - 0.3P = 75k (some other value), leading to 30k = 0.3P.

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