Decision Probability

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Decision Probability · Grade 11 · Worksheet 2

1. Charlotte is evaluating two investment strategies. Strategy A: gain $2700 with probability 0.37, lose $1200 with probability 0.63. Strategy B: gain $1700 with probability 0.62, lose $700 with probability 0.38. Which strategy has the higher expected value and by how much? Answer: Strategy B by $52 Solution: E(A) = (2700 × 0.37) + (-1200 × 0.63) = 999 + (-756) = 243 E(B) = (1700 × 0.62) + (-700 × 0.38) = 1054 + (-266) = 788 E(A) = 243, E(B) = 788 788 - 243 = 545 Since E(B) > E(A), Strategy B has the higher expected value by $545.
   Full step-by-step solution

   Step 1: Calculate expected value for Strategy A E(A) = (2700 × 0.37) + (-1200 × 0.63) = 999 + (-756) = 243 Step 2: Calculate expected value for Strategy B E(B) = (1700 × 0.62) + (-700 × 0.38) = 1054 + (-266) = 788 Step 3: Compare the expected values E(A) = 243, E(B) = 788 788 - 243 = 545 Step 4: Determine which strategy is better Since E(B) > E(A), Strategy B has the higher expected value by $545. The answer is Strategy B by $545.

2. Aroha is evaluating two investment options. Option A: gain $3200 with probability 0.30, lose $800 with probability 0.70. Option B: gain $2100 with probability 0.50, lose $600 with probability 0.50. Which option has the higher expected value and by how much? Answer: Option B by $50 Solution: E(A) = (3200 × 0.30) + (-800 × 0.70) = 960 + (-560) = 400 E(B) = (2100 × 0.50) + (-600 × 0.50) = 1050 + (-300) = 750 E(A) = 400, E(B) = 750 750 - 400 = 350 Since E(B) > E(A), Option B has the higher expected value by $350.
   Full step-by-step solution

   Step 1: Calculate expected value for Option A E(A) = (3200 × 0.30) + (-800 × 0.70) = 960 + (-560) = 400 Step 2: Calculate expected value for Option B E(B) = (2100 × 0.50) + (-600 × 0.50) = 1050 + (-300) = 750 Step 3: Compare the expected values E(A) = 400, E(B) = 750 750 - 400 = 350 Step 4: Determine which option is better Since E(B) > E(A), Option B has the higher expected value by $350. The answer is Option B by $350.

3. Mere is designing a board game where players spin a spinner divided into four equal sectors colored red, blue, green, and yellow. If a player lands on red, they win $12; on blue, they win $6; on green, they lose $4; and on yellow, they lose $10. What is the expected value of one spin? Should a player expect to gain or lose money in the long run? Answer: $1.00 Solution: There are 4 equal sectors, so the probability of landing on any one color is 1/4.
   Full step-by-step solution

   Step 1: There are 4 equal sectors, so the probability of landing on any one color is 1/4. Step 2: Expected value = (probability of red) x (win on red) + (probability of blue) x (win on blue) + (probability of green) x (loss on green) + (probability of yellow) x (loss on yellow). Step 3: Expected value = (1/4)(12) + (1/4)(6) + (1/4)(-4) + (1/4)(-10). Step 4: Calculate each term: (1/4)(12) = 3, (1/4)(6) = 1.5, (1/4)(-4) = -1, (1/4)(-10) = -2.5. Step 5: Sum: 3 + 1.5 - 1 - 2.5 = 1. The expected value is $1.00, so a player expects to gain $1 per spin in the long run.

4. Olivia is managing a food truck at a weekend festival. She must decide whether to buy 150 units of a popular ingredient at a cost of $5 per unit today, or wait until Saturday morning when a supplier offers a discounted rate of $3 per unit but only if weather conditions are favorable. If she buys today, she is guaranteed to sell all 150 units at $12 each. If she waits, there is a 60% chance of favorable weather, allowing her to buy at $3 per unit and sell at $12 each, but a 40% chance of bad weather, in which case the supplier cancels the discount and she cannot buy the ingredient at all, earning $0. Based on expected profit, should Olivia buy today or wait, and what is the expected value of the better decision? Answer: Buy today with expected profit of $1,050 Solution: Calculate profit if buying today. Cost = 150 units * $5 = $750. Revenue = 150 units * $12 = $1,800.
   Full step-by-step solution

   Step 1: Calculate profit if buying today. Cost = 150 units * $5 = $750. Revenue = 150 units * $12 = $1,800. Profit = $1,800 - $750 = $1,050. Since this is guaranteed, expected profit = $1,050. Step 2: Calculate expected profit if waiting. There are two outcomes: - Favorable weather (60% chance): Cost = 150 * $3 = $450. Revenue = $1,800. Profit = $1,800 - $450 = $1,350. - Bad weather (40% chance): Cannot buy, profit = $0. Expected profit = (0.60 * $1,350) + (0.40 * $0) = $810 + $0 = $810. Step 3: Compare expected profits: Buy today gives $1,050; wait gives $810. Since $1,050 > $810, Olivia should buy today. The expected value of the better decision (buy today) is $1,050.

5. A pharmaceutical company is testing a new drug and claims it reduces recovery time by at least 2 days compared to the standard treatment. They conduct a clinical trial with 150 patients and find a sample mean reduction of 1.8 days with a standard deviation of 0.9 days. Using a significance level of α = 0.05, test the hypothesis H₀: μ ≥ 2 against H₁: μ < 2. What is the p-value for this one-tailed test, and should the company reject the null hypothesis? Answer: p-value ≈ 0.0032, reject H₀ Solution: The company claims the drug reduces recovery time by at least 2 days compared to standard treatment. Null hypothesis: H₀: μ ≥ 2 Alternative hypothesis: H₁: μ < 2 This is a one-tailed test (left-tailed).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: State the hypotheses** The company claims the drug reduces recovery time by at least 2 days compared to standard treatment. Null hypothesis: H₀: μ ≥ 2 Alternative hypothesis: H₁: μ < 2 This is a one-tailed test (left-tailed). --- **Step 2: Identify given data** Sample size: n = 150 Sample mean: x̄ = 1.8 Population standard deviation is unknown, sample standard deviation: s = 0.9 Significance level: α = 0.05 --- **Step 3: Compute the test statistic** Since n is large (n > 30), we can use the z-test (though t-test is also fine, but with large n, z and t are very close). Test statistic formula for one sample mean: z = (x̄ - μ₀) / (s / sqrt(n)) Where μ₀ = 2 (value under H₀). z = (1.8 - 2) / (0.9 / sqrt(150)) z = (-0.2) / (0.9 / 12.2474) z = -0.2 / 0.07348 z ≈ -2.722 --- **Step 4: Find the p-value** For a left-tailed test: p-value = P(Z < -2.722) We can use the standard normal table or approximation. From z-tables: z = -2.72 → cumulative probability ≈ 0.00326 z = -2.73 → cumulative probability ≈ 0.00317 Interpolating for -2.722: p-value ≈ 0.0032 --- **Step 5: Compare p-value with α** p-value ≈ 0.0032 α = 0.05 Since p-value < α, we reject H₀. --- **Step 6: Conclusion** At α = 0.05, there is sufficient evidence to reject the null hypothesis. The data suggest that the mean reduction is less than 2 days. --- **Final answer:** p-value ≈ 0.0032, reject H₀

6. Noah is evaluating two investment options. Option A: gain $8400 with probability 0.35, lose $3200 with probability 0.65. Option B: gain $5600 with probability 0.55, lose $1800 with probability 0.45. Which option has the higher expected value and by how much? Answer: Option B by $170 Solution: E(A) = (8400 × 0.35) + (-3200 × 0.65) = 2940 + (-2080) = 860 E(B) = (5600 × 0.55) + (-1800 × 0.45) = 3080 + (-810) = 2270 E(A) = 860, E(B) = 2270 2270 - 860 = 1410 Since E(B) > E(A), Option B has the higher expected value by $1410.
   Full step-by-step solution

   Step 1: Calculate expected value for Option A E(A) = (8400 × 0.35) + (-3200 × 0.65) = 2940 + (-2080) = 860 Step 2: Calculate expected value for Option B E(B) = (5600 × 0.55) + (-1800 × 0.45) = 3080 + (-810) = 2270 Step 3: Compare the expected values E(A) = 860, E(B) = 2270 2270 - 860 = 1410 Step 4: Determine which option is better Since E(B) > E(A), Option B has the higher expected value by $1410. The answer is Option B by $1410.