The only sure way to get all questions wrong on a test is by knowing all answers.

Recently, I was thinking about the probability of getting all questions wrong on a test.

Considering a to be the number of choices each question has, and b to be the total number of questions (each question has only 1 correct answer), then the probability of getting all questions wrong is [(a-1)/a]^b.

For a normal exam of - say, 30 questions with 4 options each, the probability becomes 0.000178 or 0.0178%. This probability becomes infinitesimally small when the number of questions increases.

Makes me wonder - if answer selection is truly random, then it is highly unlikely that I get 0 questions right.

However, the more one is well-versed with the exam content, the higher the probability of knowing the correct answer is.

This means that knowing all answers implies the knowledge of all wrong answers, hence making the probability of getting all answers wrong close to 1 (given that the test-taker's intention is to get all answers wrong).

Hence, knowing everything can create a homogenous "superposition" of all different scores, with the actual score being linked to the person's intention.

As the test-taker becomes more unsure of the exam content, the probabilities of the final score shift towards a non-homogenous superposition of lower scores, hence making a lower score more probable.

At the end, my question remains - what is the fundamental force that resulted in a given grade (other than the readiness)? Clearly, it's the person's intent. Can intentions really be considered as the strongest indicator of consequence?