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Sagot :
Sure, Sebastian is looking to determine which of the provided scoring schemes is the most favorable to a test taker. Let’s carefully evaluate the expected value for each of the given scoring schemes based on the probabilities of guessing answers correctly or incorrectly.
### Definitions:
- Expected Value (EV): The average score a test taker can expect per question, based on the probabilities of answering correctly or incorrectly.
- Probability of Correct Answer (P_correct): When guessing, calculated as [tex]\( P_{correct} = \frac{1}{\text{number of choices}} \)[/tex].
- Probability of Incorrect Answer (P_incorrect): This is the complement of P_correct, so [tex]\( P_{incorrect} = 1 - P_{correct} \)[/tex].
### Scoring Scheme Calculations:
#### Scheme A (5 choices):
- Correct answer points: 1
- Incorrect answer points: [tex]\( -\frac{1}{4} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{5} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{5} = \frac{4}{5} \)[/tex]
[tex]\[ EV_A = \left(\frac{1}{5} \times 1\right) + \left(\frac{4}{5} \times -\frac{1}{4}\right) \][/tex]
[tex]\[ EV_A = \frac{1}{5} - \frac{1}{5} = 0 \][/tex]
#### Scheme B (3 choices):
- Correct answer points: 2
- Incorrect answer points: [tex]\( -\frac{1}{2} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{3} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{3} = \frac{2}{3} \)[/tex]
[tex]\[ EV_B = \left(\frac{1}{3} \times 2\right) + \left(\frac{2}{3} \times -\frac{1}{2}\right) \][/tex]
[tex]\[ EV_B = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \approx 0.3333 \][/tex]
#### Scheme C (4 choices):
- Correct answer points: 1
- Incorrect answer points: [tex]\( -\frac{1}{3} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{4} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{4} = \frac{3}{4} \)[/tex]
[tex]\[ EV_C = \left(\frac{1}{4} \times 1\right) + \left(\frac{3}{4} \times -\frac{1}{3}\right) \][/tex]
[tex]\[ EV_C = \frac{1}{4} - \frac{1}{4} = 0 \][/tex]
#### Scheme D (3 choices):
- Correct answer points: 3
- Incorrect answer points: [tex]\( -\frac{1}{3} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{3} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{3} = \frac{2}{3} \)[/tex]
[tex]\[ EV_D = \left(\frac{1}{3} \times 3\right) + \left(\frac{2}{3} \times -\frac{1}{3}\right) \][/tex]
[tex]\[ EV_D = 1 - \frac{2}{9} = \frac{7}{9} \approx 0.7778 \][/tex]
### Conclusion:
Comparing the expected values:
- [tex]\( EV_A = 0 \)[/tex]
- [tex]\( EV_B \approx 0.3333 \)[/tex]
- [tex]\( EV_C = 0 \)[/tex]
- [tex]\( EV_D \approx 0.7778 \)[/tex]
Scheme D, with an expected value of approximately 0.7778, turns out to be the most favorable for the test taker.
### Definitions:
- Expected Value (EV): The average score a test taker can expect per question, based on the probabilities of answering correctly or incorrectly.
- Probability of Correct Answer (P_correct): When guessing, calculated as [tex]\( P_{correct} = \frac{1}{\text{number of choices}} \)[/tex].
- Probability of Incorrect Answer (P_incorrect): This is the complement of P_correct, so [tex]\( P_{incorrect} = 1 - P_{correct} \)[/tex].
### Scoring Scheme Calculations:
#### Scheme A (5 choices):
- Correct answer points: 1
- Incorrect answer points: [tex]\( -\frac{1}{4} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{5} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{5} = \frac{4}{5} \)[/tex]
[tex]\[ EV_A = \left(\frac{1}{5} \times 1\right) + \left(\frac{4}{5} \times -\frac{1}{4}\right) \][/tex]
[tex]\[ EV_A = \frac{1}{5} - \frac{1}{5} = 0 \][/tex]
#### Scheme B (3 choices):
- Correct answer points: 2
- Incorrect answer points: [tex]\( -\frac{1}{2} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{3} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{3} = \frac{2}{3} \)[/tex]
[tex]\[ EV_B = \left(\frac{1}{3} \times 2\right) + \left(\frac{2}{3} \times -\frac{1}{2}\right) \][/tex]
[tex]\[ EV_B = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \approx 0.3333 \][/tex]
#### Scheme C (4 choices):
- Correct answer points: 1
- Incorrect answer points: [tex]\( -\frac{1}{3} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{4} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{4} = \frac{3}{4} \)[/tex]
[tex]\[ EV_C = \left(\frac{1}{4} \times 1\right) + \left(\frac{3}{4} \times -\frac{1}{3}\right) \][/tex]
[tex]\[ EV_C = \frac{1}{4} - \frac{1}{4} = 0 \][/tex]
#### Scheme D (3 choices):
- Correct answer points: 3
- Incorrect answer points: [tex]\( -\frac{1}{3} \)[/tex]
- [tex]\( P_{correct} = \frac{1}{3} \)[/tex]
- [tex]\( P_{incorrect} = 1 - \frac{1}{3} = \frac{2}{3} \)[/tex]
[tex]\[ EV_D = \left(\frac{1}{3} \times 3\right) + \left(\frac{2}{3} \times -\frac{1}{3}\right) \][/tex]
[tex]\[ EV_D = 1 - \frac{2}{9} = \frac{7}{9} \approx 0.7778 \][/tex]
### Conclusion:
Comparing the expected values:
- [tex]\( EV_A = 0 \)[/tex]
- [tex]\( EV_B \approx 0.3333 \)[/tex]
- [tex]\( EV_C = 0 \)[/tex]
- [tex]\( EV_D \approx 0.7778 \)[/tex]
Scheme D, with an expected value of approximately 0.7778, turns out to be the most favorable for the test taker.
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