Two students A and B use two different mechanisms of guessing answers of $n$ multiple choice questions with 4 alternatives given.
A randomly chooses the alternatives, while B constantly chooses the same alternative (for example: he bubbles the option A for all n questions on the answer sheet) if the answers of the questions are distributed evenly...can one show the efficiency of mechanism using probability ?
We want to compare A's expectation (the average number of answers he will get correct) with A's expectation. As you said, B's expectation is ${n\over 4}$ because it's simply the average number of questions with the constant answer he puts down.
For A's expectation, we use the fact that the expectation of a sum is the sum of the expectations. That is, the average number of questions that A gets right is the average number of times he gets question $1$ right, plus the average number of times he gets question $2$ right, plus ... . Now he clearly gets each question right one-quarter of the time, so his expectation is also ${n\over4}$.