Question

Up until 2015, the SAT had a "guessing penalty". Consider an exam with 40 questions each with six possible choices. Only one is correct. Each correct answer is worth +20 points and each incorrect one is worth -5 points. Consider answering each question by rolling a six-sided die for each question and simply choosing the corresponding answer.

Answer 1

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.