Strategies to guess choices for multiple choice questions Multiple choice questions (MCQs) are common in examinations over here in Singapore. A set of, say, $40$ questions are given to students, and each is accompanied with a list of $4$ choices of answer, $A, B, C$ or $D$.
  
  Suppose a student did not study for a particular test. Out of desperation, he decides to "guess" a choice for each question to secure at least some points. He considers two possible strategies:
  
  
  For each question, he picks a random choice as his answer.
  He picks a random choice, say, $B$ and uses it for all his answer.
  
  
  Assuming the correct answers are randomly distributed and his guess is completely random, which strategy would give a higher probability of securing more correct answers than the other?


Intuitively, the second strategy would give a higher probability of securing more marks. However, I am unable to come up with a mathematical proof (or disproof).

For the second strategy, we only need consider the probability that the correct answer is the choice that was picked. Under the assumption that the correct answers are randomly distributed, the probability of a particular choice being the correct answer is $\frac{1}{4}$. Hence, the student would get approximately $25\%$ of his guesses correct.

However, we can also use the same argument for the first strategy to say that the student would also get approximately $25\%$ of his guesses correct. This would imply that both strategies are equally effective, but I am pretty sure the second strategy is more effective.


EDIT: In order to prevent psychological factors from distorting reality,  I decided to write a program (in C#) that simulates the aforementioned MCQ tests. I configured the program to simulate the taking of $1000$ randomly generated MCQ tests with $40$ questions each using both strategies. 

It turns out that the percentage scores for both strategies have the same average ($\approx25\%$) and the same standard deviation ($\approx 6.84$)!

Code:

static Random rng = new Random((int)DateTime.Now.Ticks);

static void GenerateAnswers(int[] answers)
{
    for (int i = 0; i &lt; answers.Length; i++)
    {
        answers[i] = rng.Next(4);
    }
}

static int Strategy1(int[] answers)
{
    int score = 0;
    foreach (int answer in answers)
    {
        if (rng.Next(4) == answer)
        {
            score++;
        }
    }
    return score;
}

static int Strategy2(int[] answers)
{
    int choice = rng.Next(4);
    return answers.Count(x =&gt; x == choice);
}
Both will give an the same probability of securing marks. (Both will give you an expected marks of 25%.)

The key thing to note is that each question is independent of another.  This means that if the answer to the first question is B, then it does not make the answer to the second question more likely to be B (or any other choice, for that matter).

Thus, you can consider each question separately: the 25% chance of getting the first question correct does not affect the 25% chance that you'll get the second question correct, so whatever you choose for the first question isn't going to make a certain answer for the next question more correct than another.

Note: However, psychology of test setters might come into play, where test setters unknowingly avoid having consecutive answers that are both C, or other biases, and this might affect the resultant probability.
Suppose four unprepared students sit for the test, say $W,X,Y,Z$.  They decide that $Z$ will always guess $A$, $Y$ will always guess $B$, $X$ will always guess $C$, and $W$ will always guess $D$.

On any given test, it is unlikely that all four students will get the same score.  At random one of them, say $Y$, may get slightly more than 25%, while another, say $W$, will get slightly less than 25%.  Afterwards $Y$ will go on math.stackexchange with intuition that always picking $B$ is the winning strategy, because it leads to good results.  But also $W$ will go on math.stackexchange with intuition that picking at random is a better strategy, because always picking $D$ leads to bad results.

OP is $Y$.