# Expected Value on Guessing — Solution ## Problem Restatement - **Correct answer:** +7 points - **Incorrect answer:** points - **Blank:** +2 points - **How many choices must be eliminated from 5 options so that random guessing gives an expected value greater than 2?** ## Step 1: Define Variables Let: - \( n \): Total choices per question (\( n = 5 \)) - \( k \): Number of choices you can eliminate So, number of remaining choices = \( n - k \) - Probability of picking the correct answer (after eliminating \( k \) choices): \( P(\text{correct}) = \frac{1}{n - k} \) - Probability of picking an incorrect answer: \( P(\text{incorrect}) = 1 - \frac{1}{n - k} = \frac{n - k - 1}{n - k} \) ## Step 2: Calculate Expected Value of Guessing Expected value (EV) for guessing: \[ \text{EV} = P(\text{correct}) \times 7 + P(\text{incorrect}) \times \] \[ \text{EV} = \frac{1}{n - k} \times 7 + \frac{n - k - 1}{n - k} \times \] \[ \text{EV} = \frac{7}{n - k} \] ## Step 3: Compare with Leaving Blank - Leaving blank gives 2 points. - Want: \(\text{EV} > 2\) So: \[ \frac{7}{n - k} > 2 \] ## Step 4: Solve for \( k \) \[ \frac{7}{n - k} > 2 \] \[ 7 > 2(n - k) \] \[ \frac{7}{2} > n - k \] \[ n - k < 3.5 \] Since \( n - k \) must be an integer (number of choices left): - Smallest integer less than 3.5 is 3. So, \( n - k = 3 \). \[ k = n - 3 = 5 - 3 = 2 \] ## Step 5: Final Answer **You must eliminate at least 2 choices out of 5 so that guessing gives an expected value greater than 2 points.** --- ## Summary Table | Choices Left (\( n-k \)) | EV for Guessing | Advantageous? (\(>2\)) | |--------------------------|-----------------|------------------------| | 5 | 1.4 | No | | 4 | 1.75 | No | | 3 | 2.33 | **Yes** | | 2 | 3.5 | Yes | --- ## Key Takeaway **Eliminate at least 2 out of 5 choices for guessing to be better than leaving the question blank.**

# Expected Value on Guessing — Solution ## Problem Restatement - **Correct answer:** +7 points - **Incorrect answer:** 0 points - **Blank:** +2 points - **Objective:** Determine how many choices must be eliminated from 5 options for random guessing to yield an expected value greater than 2. ## Step 1: Define Variables Let: - \( n \): Total choices per question (\( n = 5 \)) - \( k \): Number of choices eliminated - Remaining choices = \( n - k \) ### Probabilities - Probability of correct answer: \[ P(\text{correct}) = \frac{1}{n - k} \] - Probability of incorrect answer: \[ P(\text{incorrect}) = 1 - P(\text{correct}) = \frac{n - k - 1}{n - k} \] ## Step 2: Calculate Expected Value of Guessing The expected value (EV) for guessing is calculated as: \[ \text{EV} = P(\text{correct}) \times 7 + P(\text{incorrect}) \times 0 \] \[ \text{EV} = \frac{1}{n - k} \times 7 + \frac{n - k - 1}{n - k} \times 0 = \frac{7}{n - k} \] ## Step 3: Compare with Leaving Blank Leaving a question blank provides 2 points. We want: \[ \text{EV} > 2 \] This leads to: \[ \frac{7}{n - k} > 2 \] ## Step 4: Solve for \( k \) 1. Rearranging gives: \[ 7 > 2(n - k) \] \[ \frac{7}{2} > n - k \] \[ n - k < 3.5 \] 2. Since \( n - k \) must be an integer, the largest integer less than 3.5 is 3. Thus: \[ n - k = 3 \] \[ k = n - 3 = 5 - 3 = 2 \] ## Step 5: Final Answer **You must eliminate at least 2 choices out of 5 so that guessing provides an expected value greater than 2 points.** ## Key Takeaway **Eliminate at least 2 out of 5 choices for guessing to be more advantageous than leaving the question blank.**