To analyze the situation of answering each SAT question by rolling a six-sided die, let's first establish the scoring mechanism based on correct and incorrect responses.
Scoring Breakdown:
- Each correct answer scores +20 points.
- Each incorrect answer scores -5 points.
- If a question is left unanswered, it scores 0 points.
When rolling a six-sided die for each question:
- The probability of rolling the correct answer is \( \frac{1}{6} \) (since there is 1 correct answer out of 6 choices).
- The probability of rolling an incorrect answer is \( \frac{5}{6} \).
Expected Score Calculation for One Question: Let the expected score \( E \) for one question be calculated as follows: \[ E = P(\text{correct}) \times \text{score for correct} + P(\text{incorrect}) \times \text{score for incorrect} \] Plugging in the probabilities and scores: \[ E = \left(\frac{1}{6} \times 20\right) + \left(\frac{5}{6} \times (-5)\right) \]
Calculating each term:
- Correct answer contribution: \( \frac{1}{6} \times 20 = \frac{20}{6} = \frac{10}{3} \)
- Incorrect answer contribution: \( \frac{5}{6} \times (-5) = -\frac{25}{6} \)
Now, summing these contributions together: \[ E = \frac{10}{3} - \frac{25}{6} \] To perform this subtraction, we need a common denominator, which is 6: \[ E = \frac{20}{6} - \frac{25}{6} = -\frac{5}{6} \]
Expected Score for All Questions: Since there are 40 questions, the total expected score \( E_{total} \) is: \[ E_{total} = 40 \times E = 40 \times \left(-\frac{5}{6}\right) = \frac{-200}{6} = -\frac{100}{3} \approx -33.33 \]
Conclusion: If you were to answer all 40 questions by randomly guessing (rolling a six-sided die for each), your expected score would be approximately -33.33 points. This is a negative score due to the penalty for incorrect answers outweighing the occasional correct answer that you might get.
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