Question: 2018 Fall Final - 23

The final exam for COMP 2804 has 25 multiple-choice questions. For each question, there are 4 possible answers, exactly one of which is correct. Michiel chooses a positive real number $z$ and uses the following marking scheme: For each correct answer, a student receives 1 mark, whereas for each incorrect answer, the student receives $-z$ marks.
Jim is one of the students and answers the 25 questions, by choosing a uniformly random answer for each question; the choices are independent of each other.
Define the random variable

• X = the number of marks that Jim recevies.

For what value of $z$ is the expected value $\mathbb{E}(X)$ equal to 0?
Hint: Use the Linearity of Expectation.