Intro

We are interested in knowing the probability that a student understands the answer and concept given that the answer selected is correct.

Formulation

Take a multiple choice exam, each question has m answer choices i.e. i, ii, iii, … , m.

Let $p =$ Probability that the student understands the concept and $1 - p =$ Probability that the student doesn’t understand the concept.

Then: $C =$ “Correct Answer” and $K =$ “Understands the answer and concept.”

Probability

So, . Why are we able to disregard C in the numerator? By the definition of the probability spaces, we note that the intersection of KC is only K since K is contained in C. That is to say that K is a subset of C.

Therefore, only the values in K will be in C for the intersection KC. Continuing on…

Application

What if we only had 4 options per question with a $p =.5$? Then,

If a student doesn’t understand the problem and receives credit,

For kicks, what happens if there are infinite options per question?

The takeaway: increasing the amount of answers per question decreases the likelihood of the examinee receiving the right answer if they do not understand the problem.