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:
-
Correct Answer:
- Probability of guessing correctly = \( \frac{1}{6} \)
- Points for a correct answer = +20 points.
-
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:
-
Correct answer contribution:
- \( \frac{1}{6} \times 20 = \frac{20}{6} \approx 3.33 \)
-
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.
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