Question

In an examination consisting of 100 multiple choice questions, each question has four choices out of which only one is correct. A student scores +1 for each correct answer and a negative mark of -1/5 for each wrong answer. The correct choices are uniformly distributed across the four choices. If an unprepared student always selects the first choice for each question, then the expected value of the student’s total score in the examination would be ............ (round off to 1 decimal place)

Hint:

For an unprepared student who guesses all answers, calculate the expected value by considering the probabilities of correct and incorrect answers and multiplying by the number of questions.

Answer 1

Correct Answer: 9.9

Step 1: Determine the probability of correct and incorrect answers.
The probability of selecting the correct answer is \( \frac{1}{4} \), and the probability of selecting the incorrect answer is \( \frac{3}{4} \).
Step 2: Expected value for each question.
For each question, the student scores:
- \( +1 \) for a correct answer with probability \( \frac{1}{4} \),
- \( -\frac{1}{5} \) for an incorrect answer with probability \( \frac{3}{4} \).
The expected value for each question is: \[ \text{Expected value} = \frac{1}{4} \times 1 + \frac{3}{4} \times \left(-\frac{1}{5}\right) = \frac{1}{4} - \frac{3}{20} = \frac{5}{20} - \frac{3}{20} = \frac{2}{20} = 0.1 \] Step 3: Expected value for 100 questions.
Since there are 100 questions, the total expected value is: \[ \text{Total expected value} = 100 \times 0.1 = 10.0 \] Step 4: Conclusion.
Thus, the expected value of the student's total score in the examination is \( \boxed{0.0} \).