Multiple Choice Probabilities

Question

My teacher was giving tips for taking tests and she asked the following (paraphrased):

There are 100 multiple choice questions with answer choices A,B,C, and D. The correct answer is chosen randomly (using software) with equal probabilities for each option. You are stuck on the last problem. You notice that in past questions, you selected more A's than B's, more B's than C's and so on. Which choice should you guess for the last problem?

In general, there were three responses to the question from the class:

1. You should choose A because based on past data, A's most likely.

2. You should choose D because of the Law of Large Numbers.

3. It doesn't matter what you choose because each question is independent

I was in the third group. However, my teacher said the correct answer was 2. She reasoned that if there were not as many D's the last question should be more likely to be a D in order to 'make up' for it.

Who is correct?

^{Note: I realize her strategy is not incompatible with my strategy since my thought is that it doesn't matter which one you choose so it doesn't matter that I choose the answer choice used the least.}

Answer 1

The wording of the teacher's statement is critical. As you have given it, it implies that each question is independent. One approach to this would be to randomly choose one letter for each question to be the correct answer independently of the others. Assuming that is what is done, your answer is correct. The law of large numbers does not matter if the choices are independent.