Expected score in a multiple-choice test

Question

A test consists of 13 multiple-choice questions, each with a choice of 4 answers where only one is correct. Suppose a student simply tries to guess all the answers. For a question answered correctly the student receives 1 point and for a question answered incorrectly she/he loses 0.25 points. Find the expected score of the student assuming she/he leaves no question unanswered.

Answer 1

Hint: Expectation is a Linear operator. $$\mathsf E(\sum_{k=1}^n X_k)~=~\sum_{k=1}^n\mathsf E( X_k)$$

So... What is the expected points they will receive for any one question?

Answer 2

Split it into $14$ disjoint events, calculate the probability of each event, multiply the probability of each event by the number of points that the student is granted on the event, and sum up the results:

$$\sum\limits_{n=0}^{13}(n\cdot1-(13-n)\cdot0.25)\cdot\binom{13}{n}\cdot\left(\frac14\right)^{n}\cdot\left(1-\frac14\right)^{13-n}$$