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 determine the expected score when answering questions on the exam by rolling a six-sided die, we need to calculate both the expected score from correct answers and the penalty from incorrect answers.

### Breakdown of Scoring:
1. **Correct Answer:**
- Probability of guessing correctly = \( \frac{1}{6} \)
- Points for a correct answer = +20 points.

2. **Incorrect Answer:**
- Probability of guessing incorrectly = \( \frac{5}{6} \)
- Points for an incorrect answer = -5 points.

### Expected Score Calculation per Question:
We can calculate the expected score \( E \) for answering a single question as follows:

\[
E = ( \text{Probability of Correct} \times \text{Points for Correct} ) + ( \text{Probability of Incorrect} \times \text{Points for Incorrect} )
\]

Substituting in the values:

\[
E = \left(\frac{1}{6} \times 20\right) + \left(\frac{5}{6} \times -5\right)
\]

Calculating each term:

1. **Correct answer contribution:**
- \( \frac{1}{6} \times 20 = \frac{20}{6} \approx 3.33 \)

2. **Incorrect answer contribution:**
- \( \frac{5}{6} \times -5 = -\frac{25}{6} \approx -4.17 \)

Now combine both contributions:

\[
E = \frac{20}{6} - \frac{25}{6} = \frac{20 - 25}{6} = \frac{-5}{6} \approx -0.83
\]

### Total Expected Score for 40 Questions:
Since there are 40 questions, the total expected score \( E_{total} \) is:

\[
E_{total} = 40 \times E \approx 40 \times -0.83 \approx -33.33
\]

### Conclusion:
If a student answers all 40 questions by rolling a six-sided die for each question, the expected total score on this exam would be approximately **-33.33 points**. This negative score indicates that answering by guessing without knowledge of the material would be statistically disadvantageous due to the penalization for incorrect answers.