Solution: 2018 Fall Final - 23

Author: Michiel Smid

Question

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.

(a)

$z = 1/2$

(b)

$z = 3/4$

(c)

$z = 1/4$

(d)

$z = 1/3$

Solution

How does Linearity of Expectation apply to the question?

Jim answers 25 questions randomly.
For each question:
- With probability $P(\text{correct}) = \frac{1}{4}$, Jim gets +1 mark.
- With probability $P(\text{incorrect}) = \frac{3}{4}$, Jim loses $-z$ marks.

Let $X_i$ represent the score Jim receives for the i-th question. Let $X$ represent Jim’s total score. We can represent his final total score as: $X = \sum_{i=1}^{25} X_i$. The expected value of Jim’s total score is: $E(X) = E(\sum_{i=1}^{25} X_i)$.

We can use linearity of expectation to simplify the expression above. Using linearity of expectation: $E(X) = E(\sum_{i=1}^{25} X_i) = \sum_{i=1}^{25} E(X_i)$. Now, all we have to do is calculate $E(X_i)$ instead of calculating the expected value of the entire summation itself.

We can simplify the above expression even further. Since every question is identical and independent: $E(X) = 25 \cdot E(X_i)$ For the total expected value to be 0, we need to find the $z$ value where: $E(X_i) = 0$

Calculate the Expected Value

Using the definition of expected value: $E(X_i) = (1) \cdot P(\text{correct}) + (-z) \cdot P(\text{incorrect})$

Substituting the probabilities: $E(X_i) = (1) \cdot \frac{1}{4} + (-z) \cdot \frac{3}{4}$

Simplify: $E(X_i) = \frac{1}{4} - \frac{3z}{4}$

Set $E(X_i) = 0$: $\frac{1}{4} - \frac{3z}{4} = 0$

Rearrange the equation: $\frac{1}{4} = \frac{3z}{4}$

Cancel the common factor of $\frac{1}{4}$ by multiplying both sides by $4$: $1 = 3z$

Solve for the unknown variable $z$: $z = \frac{1}{3}$