Decision Probability

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Decision Probability · Grade 11 · Worksheet 3

1. Charlotte is a video game developer deciding between two monetization strategies for her new mobile game. In Strategy A, players pay a flat fee of $12 to download the game, with a 0.25 probability that a player will make an additional in-app purchase averaging $20, a 0.45 probability of a smaller purchase averaging $8, and a 0.30 probability of no further purchase. In Strategy B, the game is free to download, but players are shown ads; each player generates $3.50 in ad revenue, but there is a 0.15 probability that the ads will drive the player away before they generate any revenue, a 0.60 probability they stay and generate the $3.50, and a 0.25 probability they become a premium subscriber paying $15. Based on expected revenue per player, which strategy should Charlotte choose, and what is the expected revenue for that strategy? Answer: Strategy A with expected revenue of $20.60 Solution: Calculate expected revenue for Strategy A. The flat fee of $12 is guaranteed for every player.
   Full step-by-step solution

   Step 1: Calculate expected revenue for Strategy A. The flat fee of $12 is guaranteed for every player. Additional in-app purchases: with probability 0.25, a player spends $20; with probability 0.45, a player spends $8; with probability 0.30, a player spends $0. Expected additional revenue = (0.25 * 20) + (0.45 * 8) + (0.30 * 0) = 5 + 3.6 + 0 = $8.60. Total expected revenue per player = $12 + $8.60 = $20.60. Step 2: Calculate expected revenue for Strategy B. With probability 0.15, the player generates $0 (driven away). With probability 0.60, the player generates $3.50 (ad revenue only). With probability 0.25, the player generates $15 (premium subscriber). Expected revenue = (0.15 * 0) + (0.60 * 3.50) + (0.25 * 15) = 0 + 2.10 + 3.75 = $5.85. Step 3: Compare: Strategy A gives $20.60 per player, Strategy B gives $5.85 per player. Strategy A has higher expected revenue. The answer is Strategy A with expected revenue of $20.60.

2. Matiu is a data analyst for a logistics company. He is evaluating two different shipping strategies for delivering packages to a remote region. Strategy A uses a single large truck that has a 15% chance of being delayed by weather, resulting in a $1,200 penalty for late delivery, and an 85% chance of arriving on time with no penalty. The base cost of operating this truck is $4,500. Strategy B uses two smaller vans. Van 1 has a 10% chance of a mechanical issue costing $800 in repairs and a 90% chance of running smoothly with no extra cost. Van 2 has a 25% chance of taking a wrong route costing $600 in extra fuel and a 75% chance of taking the correct route with no extra cost. The base cost of operating both vans together is $3,200. Matiu must choose the strategy with the lower total expected cost (base cost plus expected penalties/repairs/fuel). Which strategy should Matiu choose, and what is its total expected cost? Answer: Strategy B, $3,500 Solution: Calculate expected total cost for Strategy A. Base cost = $4,500. Weather delay penalty = $1,200 with probability 0.15.
   Full step-by-step solution

   Step 1: Calculate expected total cost for Strategy A. Base cost = $4,500. Weather delay penalty = $1,200 with probability 0.15. No delay penalty = $0 with probability 0.85. Expected total cost = 4500 + (0.15 * 1200) + (0.85 * 0) = 4500 + 180 + 0 = $4,680. Step 2: Calculate expected total cost for Strategy B. Base cost = $3,200. For van 1: mechanical issue cost = $800 with probability 0.10, no issue cost = $0 with probability 0.90. For van 2: wrong route cost = $600 with probability 0.25, correct route cost = $0 with probability 0.75. Since the vans operate independently, there are four possible combined scenarios: - Scenario 1: Van 1 issue AND van 2 wrong route. Probability = 0.10 * 0.25 = 0.025. Total cost = 3200 + 800 + 600 = $4,600. - Scenario 2: Van 1 issue AND van 2 correct route. Probability = 0.10 * 0.75 = 0.075. Total cost = 3200 + 800 + 0 = $4,000. - Scenario 3: Van 1 no issue AND van 2 wrong route. Probability = 0.90 * 0.25 = 0.225. Total cost = 3200 + 0 + 600 = $3,800. - Scenario 4: Van 1 no issue AND van 2 correct route. Probability = 0.90 * 0.75 = 0.675. Total cost = 3200 + 0 + 0 = $3,200. Step 3: Calculate expected total cost for Strategy B: (0.025 * 4600) + (0.075 * 4000) + (0.225 * 3800) + (0.675 * 3200) = 115 + 300 + 855 + 2160 = $3,430. Step 4: Compare: Strategy A expected cost = $4,680. Strategy B expected cost = $3,430. Strategy B has the lower expected cost. The answer is Strategy B, $3,430.

3. Emma is deciding between two investment options for a $5000 bonus she received. Option A is a low-risk bond that guarantees a profit of $250 with probability 0.90, but has a 0.10 probability of losing $50 due to early withdrawal penalties. Option B is a high-risk stock that has a 0.35 probability of gaining $600, a 0.50 probability of gaining $100, and a 0.15 probability of losing $300. Which investment option has the higher expected monetary value, and what is that expected value? Answer: Option A with expected value of $220 Solution: Calculate the expected value for Option A. The outcomes are a gain of $250 (probability 0.90) and a loss of $50 (probability 0.10). Expected value A = (0.90 * 250) + (0.10 * -50) = 225 + (-5) = $220.
   Full step-by-step solution

   Step 1: Calculate the expected value for Option A. The outcomes are a gain of $250 (probability 0.90) and a loss of $50 (probability 0.10). Expected value A = (0.90 * 250) + (0.10 * -50) = 225 + (-5) = $220. Step 2: Calculate the expected value for Option B. The outcomes are a gain of $600 (probability 0.35), a gain of $100 (probability 0.50), and a loss of $300 (probability 0.15). Expected value B = (0.35 * 600) + (0.50 * 100) + (0.15 * -300) = 210 + 50 + (-45) = $215. Step 3: Compare the expected values: $220 (Option A) vs $215 (Option B). Since $220 > $215, Option A has the higher expected value. The answer is Option A with expected value of $220.

4. Sophia is deciding between two investment options. Option A: gain $160 with probability 0.6, lose $40 with probability 0.4. Option B: gain $110 with probability 0.7, lose $60 with probability 0.3. Which option has the higher expected value and by how much? Answer: Option B by $21 Solution: E(A) = (160 × 0.6) + (-40 × 0.4) = 96 + (-16) = 80 E(B) = (110 × 0.7) + (-60 × 0.3) = 77 + (-18) = 59 E(A) = 80, E(B) = 59 80 - 59 = 21 Since E(A) > E(B), Option A has the higher expected value by $21 The answer is Option A by $21.
   Full step-by-step solution

   Step 1: Calculate expected value for Option A E(A) = (160 × 0.6) + (-40 × 0.4) = 96 + (-16) = 80 Step 2: Calculate expected value for Option B E(B) = (110 × 0.7) + (-60 × 0.3) = 77 + (-18) = 59 Step 3: Compare the expected values E(A) = 80, E(B) = 59 80 - 59 = 21 Step 4: Determine which option is better Since E(A) > E(B), Option A has the higher expected value by $21 The answer is Option A by $21.