1977 A-Level H2 Mathematics

A test consists of five multiple choice questions to each of which three answers are given, only one of which is correct. For each correct answer a candidate gets $2$ marks, but loses $1$ mark for each incorrect answer. A particular candidate answers all question purely by guesswork. Draw up a table to show all the possible total marks from $-5$ to $10$ and the probability associated with each mark for this candidate. Use the table to show that the expected mark is zero, and find the variance of this distribution.