Question

On a multiple-choice test, a student is given five possible answers for each question. The student receives 1 point for a correct answer and loses 1/4 point for an incorrect answer. If the student has no idea of the correct answer for a particular question and merely guesses, what is the students expected gain or loss on the question?

Answer 1

To calculate the expected gain or loss for a student who randomly guesses on a multiple-choice question with 5 possible answers, we first need to determine the probabilities of a correct answer and an incorrect answer.

1. **Probability of a correct answer:**
Since there is only one correct answer among the five choices, the probability \( P(\text{correct}) \) is:
\[
P(\text{correct}) = \frac{1}{5}
\]

2. **Probability of an incorrect answer:**
The probability of getting an incorrect answer is:
\[
P(\text{incorrect}) = 1 - P(\text{correct}) = 1 - \frac{1}{5} = \frac{4}{5}
\]

3. **Points for a correct and incorrect answer:**
- If the student answers correctly, they gain 1 point.
- If the student answers incorrectly, they lose \( \frac{1}{4} \) point (which is the same as -0.25 points).

4. **Calculating the expected value:**
The expected value (or expected gain/loss) can be calculated as follows:
\[
\text{Expected value} = P(\text{correct}) \times (\text{points for correct}) + P(\text{incorrect}) \times (\text{points for incorrect})
\]
Plugging in the values, we get:
\[
\text{Expected value} = \left(\frac{1}{5} \times 1\right) + \left(\frac{4}{5} \times -\frac{1}{4}\right)
\]
This simplifies to:
\[
= \frac{1}{5} + \frac{4}{5} \times -\frac{1}{4}
\]
\[
= \frac{1}{5} - \frac{4}{20}
\]
\[
= \frac{1}{5} - \frac{1}{5}
\]
\[
= 0
\]

Thus, the expected gain or loss on this question is:

\[
\boxed{0}
\]