Decision Probability

lessonbunny.com

Answer Key & Explanations

Decision Probability · Grade 11 · Worksheet 1

1. Hana is an investment analyst evaluating two portfolio options for a client. Option A is a fixed-income bond that guarantees a return of $4,200 with certainty. Option B is a tech stock portfolio with variable returns: there is a 0.18 probability of a high return of $12,800, a 0.42 probability of a moderate return of $5,600, and a 0.40 probability of a low return of $1,200. Based on the expected monetary value, which option should Hana recommend to her client, and what is the expected value of that option? Answer: Option A with expected value of $4,200 Solution: Calculate the expected value for Option B. Multiply each return by its probability: (0.18 * 12800) = 2304, (0.42 * 5600) = 2352, (0.40 * 1200) = 480. Sum these: 2304 + 2352 + 480 = 5136.
   Full step-by-step solution

   Step 1: Calculate the expected value for Option B. Multiply each return by its probability: (0.18 * 12800) = 2304, (0.42 * 5600) = 2352, (0.40 * 1200) = 480. Sum these: 2304 + 2352 + 480 = 5136. So the expected value for Option B is $5,136. Step 2: Compare the expected values. Option A has a guaranteed return of $4,200. Option B has an expected return of $5,136. Step 3: Since $5,136 is greater than $4,200, Option B has the higher expected monetary value. The answer is Option B with expected value of $5,136.

2. Noah is a game designer creating a new carnival game. In the game, a player pays $6 to roll a fair 20-sided die numbered 1 through 20. If the number rolled is 1, the player wins $21. If the number rolled is a multiple of 6 (6, 12, or 18), the player wins $11. If the number rolled is a prime number greater than 10 (11, 13, 17, or 19), the player wins $16. For any other outcome, the player wins nothing. Calculate the expected value of the profit for Noah per game. Based on this expected value, should Noah keep the game as is, or must he adjust the prize structure to avoid an expected loss? Answer: -$0.35 Solution: Determine the probabilities for each winning outcome. The die has 20 faces, each with probability 1/20. Rolling a 1: probability = 1/20.
   Full step-by-step solution

   Step 1: Determine the probabilities for each winning outcome. The die has 20 faces, each with probability 1/20. Rolling a 1: probability = 1/20. Rolling a multiple of 6 (6, 12, 18): 3 outcomes, probability = 3/20. Rolling a prime greater than 10 (11, 13, 17, 19): 4 outcomes, probability = 4/20 = 1/5. All other rolls: 20 - (1 + 3 + 4) = 12 outcomes, probability = 12/20 = 3/5, and win $0. Step 2: Calculate the expected payout to the player. Expected payout = (1/20 * 21) + (3/20 * 11) + (4/20 * 16) + (12/20 * 0) = (21/20) + (33/20) + (64/20) + 0 = (21 + 33 + 64)/20 = 118/20 = 5.9. So the expected payout is $5.90. Step 3: Calculate Noah's expected profit per game. Profit = player's fee - expected payout = $6 - $5.90 = $0.10. Step 4: Since the expected profit is positive ($0.10 per game), Noah is making a profit on average. Therefore, the game is profitable as is, and he does not need to adjust the prize structure. The answer is $0.10.

3. Liam is evaluating two investment strategies. Strategy A: gain $3500 with probability 0.25, lose $900 with probability 0.75. Strategy B: gain $2100 with probability 0.45, lose $500 with probability 0.55. Which strategy has the higher expected value and by how much? Answer: Strategy B by $95 Solution: E(A) = (3500 × 0.25) + (-900 × 0.75) = 875 + (-675) = 200 E(B) = (2100 × 0.45) + (-500 × 0.55) = 945 + (-275) = 670 E(A) = 200, E(B) = 670 Since 670 > 200, Strategy B has the higher expected value.
   Full step-by-step solution

   Step 1: Calculate expected value for Strategy A E(A) = (3500 × 0.25) + (-900 × 0.75) = 875 + (-675) = 200 Step 2: Calculate expected value for Strategy B E(B) = (2100 × 0.45) + (-500 × 0.55) = 945 + (-275) = 670 Step 3: Compare the expected values E(A) = 200, E(B) = 670 Since 670 > 200, Strategy B has the higher expected value. Difference = 670 - 200 = 470 The answer is Strategy B by $470.

4. A pharmaceutical company is testing a new drug and claims it reduces recovery time by at least 2 days compared to the standard treatment. In a clinical trial with 120 patients, the mean difference in recovery time is 1.8 days with a standard deviation of 0.9 days. The researchers conduct a one-tailed hypothesis test at α = 0.05. Calculate the test statistic and determine whether there is sufficient evidence to reject the null hypothesis that the mean difference is 2 days. Answer: z = -2.43, reject H₀ Solution: In hypothesis testing, we evaluate whether sample data provides sufficient evidence against a null hypothesis. The test statistic measures how many standard errors the sample result is from the hypothesized value.
   Full step-by-step solution

   In hypothesis testing, we evaluate whether sample data provides sufficient evidence against a null hypothesis. The test statistic measures how many standard errors the sample result is from the hypothesized value. For large samples, we typically use the z-distribution. A one-tailed test focuses on evidence in one specific direction, which affects where we place the critical region for rejection.

5. Kaia is a logistics manager for a shipping company deciding between two routing strategies for a fleet of delivery vehicles. In Strategy X, she uses a central hub system. There is a 55% chance that fuel costs will be $7,000 and a 45% chance that fuel costs will be $11,000 due to variable traffic patterns. In Strategy Y, she uses a direct routing system. There is a 25% chance that fuel costs will be $5,000, a 35% chance they will be $9,000, and a 40% chance they will be $13,000. Based on the expected fuel cost, which strategy should Kaia choose to minimize expected cost, and what is that expected cost? Answer: Strategy X with expected cost of $8,800 Solution: Calculate expected fuel cost for Strategy X. Possible costs: $7,000 with probability 0.55, $11,000 with probability 0.45. Expected cost = (0.55 * 7000) + (0.45 * 11000) = 3850 + 4950 = $8,800.
   Full step-by-step solution

   Step 1: Calculate expected fuel cost for Strategy X. Possible costs: $7,000 with probability 0.55, $11,000 with probability 0.45. Expected cost = (0.55 * 7000) + (0.45 * 11000) = 3850 + 4950 = $8,800. Step 2: Calculate expected fuel cost for Strategy Y. Possible costs: $5,000 with probability 0.25, $9,000 with probability 0.35, $13,000 with probability 0.40. Expected cost = (0.25 * 5000) + (0.35 * 9000) + (0.40 * 13000) = 1250 + 3150 + 5200 = $9,600. Step 3: Compare expected costs: Strategy X = $8,800, Strategy Y = $9,600. Since $8,800 < $9,600, Strategy X has the lower expected cost. The answer is Strategy X with expected cost of $8,800.