# Expected score in a multiple-choice test

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.

• Could anybody give me a hint?
– Mary
Oct 2, 2016 at 13:53
• Yes: 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. Oct 2, 2016 at 14:26

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:
$$\sum\limits_{n=0}^{13}(n\cdot1-(13-n)\cdot0.25)\cdot\binom{13}{n}\cdot\left(\frac14\right)^{n}\cdot\left(1-\frac14\right)^{13-n}$$
Hint: Expectation is a Linear operator. $$\mathsf E(\sum_{k=1}^n X_k)~=~\sum_{k=1}^n\mathsf E( X_k)$$