# Problem Restatement - **Test Scoring:** - Correct answer: **7** points - Incorrect answer: **** points - Blank answer: **2** points - Each question has **5 choices** (A, B, C, D, E). - **Objective:** How many choices must you eliminate so that **guessing** is better than leaving it blank (i.e., expected value of guessing is **greater than 2**)? --- # Step-by-Step Solution ## 1. Define Variables Let: - \( n \) = number of choices **remaining** after elimination. - \( m \) = number of choices **eliminated**. - Total choices is **5**: \( n + m = 5 \). ## 2. Probability Breakdown - If you guess, you pick randomly among \( n \) remaining options. - Probability of correct answer: \( \frac{1}{n} \) - Probability of incorrect answer: \( 1 - \frac{1}{n} = \frac{n-1}{n} \) ## 3. Expected Value (EV) of Guessing \[ \text{EV} = \left(\frac{1}{n}\right) \cdot 7 + \left(\frac{n-1}{n}\right) \cdot = \frac{7}{n} \] ## 4. When is Guessing Advantageous? Set EV greater than 2 (the value for leaving blank): \[ \frac{7}{n} > 2 \] \[ 7 > 2n \] \[ n < 3.5 \] \( n \) must be a whole number, so possible values: \( n = 3, 2, 1 \). ## 5. How Many Choices to Eliminate? \[ n = 5 - m \] We want \( n \leq 3 \): \[ 5 - m \leq 3 \implies m \geq 2 \] So you must eliminate **at least 2 choices**. --- # Final Answer ## **You must eliminate at least 2 out of the 5 choices (so only 3 remain) for guessing to have an expected value greater than 2, making it advantageous compared to leaving the question blank.**

# Detailed Explanation of Expected Value in Guessing ## Problem Overview In a multiple-choice test with the following scoring system: - **Correct answer:** 7 points - **Incorrect answer:** 0 points - **Blank answer:** 2 points The task is to determine how many answer choices can be eliminated so that guessing is statistically better than leaving the answer blank. ## Concepts and Definitions ### Expected Value (EV) The Expected Value (EV) is a calculated average outcome of a random variable, representing the anticipated value of a variable based on its probabilities. ### Scenario Setup - Total choices available: **5** (A, B, C, D, E) - Let \( n \) be the number of choices remaining after elimination. - Let \( m \) be the number of choices eliminated. Thus: \[ n + m = 5 \] ## Analysis of Guessing ### Probability of Outcomes When guessing from \( n \) remaining options: - Probability of selecting the correct answer: \[ P(\text{correct}) = \frac{1}{n} \] - Probability of selecting an incorrect answer: \[ P(\text{incorrect}) = 1 - P(\text{correct}) = \frac{n-1}{n} \] ### Expected Value Calculation for Guessing The EV when guessing can be calculated as follows: \[ \text{EV} = P(\text{correct}) \cdot \text{points for correct} + P(\text{incorrect}) \cdot \text{points for incorrect} \] Substituting the known values: \[ \text{EV} = \left(\frac{1}{n}\right) \cdot 7 + \left(\frac{n-1}{n}\right) \cdot 0 = \frac{7}{n} \] ## Condition for Guessing to be Advantageous To determine under what conditions guessing is better than leaving the answer blank (which scores 2 points), we set up the inequality: \[ \frac{7}{n} > 2 \] ### Solving the Inequality 1. Multiply both sides by \( n \) (assuming \( n > 0 \)): \[ 7 > 2n \] 2. Rearranging gives us: \[ n < 3.5 \] Since \( n \) must be a whole number, the possible values for \( n \) are: - \( n = 3 \) - \( n = 2 \) - \( n = 1 \) ## Determining Choices to Eliminate Since \( n + m = 5 \): - To have \( n \leq 3 \) (the maximum value for which guessing is advantageous): \[ 5 - m \leq 3 \] Rearranging yields: \[ m \geq 2 \] ### Conclusion To ensure that guessing yields an expected value greater than leaving the answer blank, at least **2 choices must be eliminated** from the original set of 5. ### Final Answer **You must eliminate at least 2 out of the 5 choices (to leave only 3 remaining) for guessing to have an expected value greater than 2, making it advantageous compared to leaving the question blank.**