# Expected Value on Guessing: Elimination Strategy Let's break down the problem: - **Correct answer:** +7 points - **Incorrect answer:** points - **Blank:** +2 points - **Total choices per question:** 5 - **Goal:** For random guessing to be **better than leaving blank** (expected value \(>2\)), how many choices must you eliminate first? --- ## 1. Set Up: Probability After Eliminating Choices If you eliminate \( n \) choices, then you have \( 5 - n \) choices left. There's only one correct answer. - **Probability of guessing correctly:** \( P(\text{correct}) = \frac{1}{5 - n} \) - **Probability of guessing incorrectly:** \( P(\text{incorrect}) = \frac{5 - n - 1}{5 - n} = \frac{4 - n}{5 - n} \) --- ## 2. Expected Value of Guessing \[ E = (\text{probability correct}) \times 7 + (\text{probability incorrect}) \times \] \[ E = \frac{1}{5 - n} \times 7 + \frac{4 - n}{5 - n} \times = \frac{7}{5 - n} \] --- ## 3. Set Inequality: Expected Value > 2 \[ \frac{7}{5 - n} > 2 \] Solve for \( n \): \[ 7 > 2(5 - n) \] \[ 7 > 10 - 2n \] \[ 2n > 10 - 7 \] \[ 2n > 3 \] \[ n > 1.5 \] --- ## 4. Interpretation Since \( n \) is the number of choices eliminated and must be a **whole number**, you need to eliminate at least **2 choices**. --- ## 5. Final Answer **You must eliminate at least 2 choices (leaving 3 options) for random guessing to have an expected value higher than leaving the question blank.** --- ### Summary Table | Choices Eliminated (\( n \)) | Choices Left | \( E = \frac{7}{5-n} \) | \( E > 2 \)? | |------------------------------|--------------|--------------------------|--------------| | | 5 | 1.4 | No | | 1 | 4 | 1.75 | No | | 2 | 3 | 2.33 | **Yes** | | 3 | 2 | 3.5 | Yes | | 4 | 1 | 7 | Yes | **[Image: Table showing expected values for different numbers of choices eliminated]** --- ## Key Takeaway > **Eliminate at least 2 choices to make guessing advantageous (\( E > 2 \)).**

# Expected Value on Guessing: Elimination Strategy Let's analyze the scenario where you want to determine when guessing answers on a test becomes advantageous. ## 1. Understanding the Scoring System - **Correct answer:** +7 points - **Incorrect answer:** 0 points - **Blank answer:** +2 points - **Total choices per question:** 5 - **Objective:** Determine how many choices must be eliminated so that the expected value of guessing exceeds 2 points. --- ## 2. Probability After Eliminating Choices If \( n \) choices are eliminated, the number of remaining choices is \( 5 - n \). - **Probability of guessing correctly:** \[ P(\text{correct}) = \frac{1}{5 - n} \] - **Probability of guessing incorrectly:** \[ P(\text{incorrect}) = \frac{4 - n}{5 - n} \] --- ## 3. Expected Value Calculation The expected value \( E \) for guessing can be calculated as follows: \[ E = P(\text{correct}) \times 7 + P(\text{incorrect}) \times 0 \] \[ E = \frac{1}{5 - n} \times 7 + \frac{4 - n}{5 - n} \times 0 = \frac{7}{5 - n} \] --- ## 4. Setting the Inequality To find when the expected value is greater than 2, we set up the following inequality: \[ \frac{7}{5 - n} > 2 \] ### Solving the Inequality 1. Multiply both sides by \( 5 - n \) (assuming \( 5 - n > 0 \)): \[ 7 > 2(5 - n) \] 2. Expand and rearrange: \[ 7 > 10 - 2n \] \[ 2n > 3 \] \[ n > 1.5 \] --- ## 5. Conclusion Since \( n \) must be a whole number, you need to eliminate at least **2 choices** for guessing to become advantageous. ### Final Answer **You must eliminate at least 2 choices (leaving 3 options) to ensure that the expected value of guessing is greater than leaving the question blank.** --- ## Key Takeaway > **Eliminate at least 2 choices to make guessing advantageous (\( E > 2 \)).**