Decision Probability

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Decision Probability · Grade 11 · Worksheet 1

1. A pharmaceutical company is testing a new drug and claims it reduces recovery time by at least 2 days compared to the standard treatment. The null hypothesis states H₀: μ = 2 (mean reduction is 2 days), while the alternative hypothesis states H₁: μ < 2 (mean reduction is less than 2 days). After conducting the trial with a sample of patients, researchers calculate a p-value of 0.032. Using a significance level of α = 0.05, should the company reject the null hypothesis and conclude the drug is less effective than claimed? Answer: B. yes Solution: The null hypothesis is H₀: μ = 2 (mean reduction is 2 days). The alternative hypothesis is H₁: μ < 2 (mean reduction is less than 2 days). This is a one-tailed test in the left direction.
   Full step-by-step solution

   Step 1: State the hypotheses The null hypothesis is H₀: μ = 2 (mean reduction is 2 days). The alternative hypothesis is H₁: μ < 2 (mean reduction is less than 2 days). This is a one-tailed test in the left direction. Step 2: Identify the significance level The significance level α is given as 0.05. Step 3: Compare the p-value with α The p-value from the sample data is 0.032. We compare: p-value = 0.032 and α = 0.05. Since 0.032 is less than 0.05, the p-value is less than α. Step 4: Decision rule If p-value ≤ α, reject H₀. If p-value > α, fail to reject H₀. Here, p-value (0.032) ≤ α (0.05), so we reject H₀. Step 5: Conclusion Rejecting H₀ means there is sufficient evidence to support the alternative hypothesis H₁: μ < 2. Thus, the drug is less effective than claimed — the mean reduction is less than 2 days. Final answer: Yes

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. 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 triangular region is bounded by the parabola y = 4 - x² and the x-axis. Calculate the exact area of this region using integration methods. Answer: 32/3 Solution: Find the x-intercepts of the parabola by setting y = 0: 4 - x² = 0 → x² = 4 → x = -2 and x = 2. The area under the curve from x = -2 to x = 2 is given by the definite integral: ∫ from -2 to 2 of (4 - x²) dx.
   Full step-by-step solution

   Step 1: Find the x-intercepts of the parabola by setting y = 0: 4 - x² = 0 → x² = 4 → x = -2 and x = 2. Step 2: The area under the curve from x = -2 to x = 2 is given by the definite integral: ∫ from -2 to 2 of (4 - x²) dx. Step 3: Find the antiderivative: The antiderivative of 4 is 4x, and the antiderivative of x² is x³/3. So the antiderivative of (4 - x²) is 4x - x³/3. Step 4: Evaluate from -2 to 2: [4(2) - (2)³/3] - [4(-2) - (-2)³/3] = [8 - 8/3] - [-8 - (-8/3)] = [8 - 8/3] - [-8 + 8/3]. Step 5: Simplify: (24/3 - 8/3) - (-24/3 + 8/3) = (16/3) - (-16/3) = 16/3 + 16/3 = 32/3. The exact area is 32/3.

5. Liam is evaluating two investment portfolios. Portfolio X: gain $3100 with probability 0.41, lose $900 with probability 0.59. Portfolio Y: gain $2100 with probability 0.61, lose $700 with probability 0.39. Which portfolio has the higher expected value and by how much? Answer: Portfolio Y by $143 Solution: E(X) = (3100 × 0.41) + (-900 × 0.59) E(X) = 1271 + (-531) E(X) = 740 E(Y) = (2100 × 0.61) + (-700 × 0.39) E(Y) = 1281 + (-273) E(Y) = 1008 E(X) = 740, E(Y) = 1008 1008 - 740 = 268 Since E(Y) > E(X), Portfolio Y has the higher expected value by $268.
   Full step-by-step solution

   Step 1: Calculate expected value for Portfolio X E(X) = (3100 × 0.41) + (-900 × 0.59) E(X) = 1271 + (-531) E(X) = 740 Step 2: Calculate expected value for Portfolio Y E(Y) = (2100 × 0.61) + (-700 × 0.39) E(Y) = 1281 + (-273) E(Y) = 1008 Step 3: Compare the expected values E(X) = 740, E(Y) = 1008 1008 - 740 = 268 Step 4: Determine which portfolio is better Since E(Y) > E(X), Portfolio Y has the higher expected value by $268. The answer is Portfolio Y by $268.

6. P(X ≥ 2) where X ~ Binomial(n=5, p=0.4) = ? Answer: 0.66304 Solution: We are given: X ~ Binomial(n = 5, p = 0.4) We want: P(X ≥ 2) P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where C(n, k) = n! / (k! (n-k)!) Here n = 5, p = 0.4, 1-p = 0.6.
   Full step-by-step solution

   We are given: X ~ Binomial(n = 5, p = 0.4) We want: P(X ≥ 2) Step 1: Recall that for a binomial distribution, P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where C(n, k) = n! / (k! (n-k)!) Here n = 5, p = 0.4, 1-p = 0.6. Step 2: Instead of computing P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) directly, we can use the complement rule: P(X ≥ 2) = 1 - P(X < 2) = 1 - [P(X = 0) + P(X = 1)]. Step 3: Compute P(X = 0): C(5, 0) = 1 P(X = 0) = 1 * (0.4)^0 * (0.6)^5 = 1 * 1 * (0.6)^5 0.6^5 = 0.6 * 0.6 = 0.36 0.36 * 0.36 = 0.1296 0.1296 * 0.6 = 0.07776 So P(X = 0) = 0.07776. Step 4: Compute P(X = 1): C(5, 1) = 5 P(X = 1) = 5 * (0.4)^1 * (0.6)^4 0.4^1 = 0.4 0.6^4 = 0.6^2 * 0.6^2 = 0.36 * 0.36 = 0.1296 So P(X = 1) = 5 * 0.4 * 0.1296 First 5 * 0.4 = 2.0 2.0 * 0.1296 = 0.2592. Step 5: Sum P(X = 0) + P(X = 1): 0.07776 + 0.2592 = 0.33696. Step 6: Apply complement rule: P(X ≥ 2) = 1 - 0.33696 = 0.66304. Final answer: 0.66304

7. Noah is evaluating two investment options. Option A: gain $3160 with probability 0.41, lose $610 with probability 0.59. Option B: gain $2160 with probability 0.61, lose $710 with probability 0.39. Which option has the higher expected value and by how much? Answer: Option A by $86.00 Solution: E(A) = (3160 × 0.41) + (-610 × 0.59) = 1295.60 + (-359.90) = 935.70 E(B) = (2160 × 0.61) + (-710 × 0.39) = 1317.60 + (-276.90) = 1040.70 E(A) = 935.70, E(B) = 1040.70 Since 1040.70 > 935.70, Option B has the higher expected value.
   Full step-by-step solution

   Step 1: Calculate expected value for Option A E(A) = (3160 × 0.41) + (-610 × 0.59) = 1295.60 + (-359.90) = 935.70 Step 2: Calculate expected value for Option B E(B) = (2160 × 0.61) + (-710 × 0.39) = 1317.60 + (-276.90) = 1040.70 Step 3: Compare the expected values E(A) = 935.70, E(B) = 1040.70 Since 1040.70 > 935.70, Option B has the higher expected value. Difference = 1040.70 - 935.70 = 105.00 The answer is Option B by $105.00.