A multiple choice exam has 4 choices for each question. A student has studied enough so that the probability they will know the answer to a question is 0.5, the probability that they will be able to eliminate one choice is 0.25, otherwise all 4 choices seem equally plausible. If they know the answer they will get the question right. If not, they have to guess from the 3 or 4 choices. As the teacher, you want to test to measure what the student knows. If the student answers a question correctly, what is the probability they knew the answer?

probability that the student knew the answer given that he answered the question correctly is 0.7742 (77.42%)

Step-by-step explanation:

a student can get the question right in 3 ways:

- knowing the answer with probability 0.5

- eliminating one of the 4 choices and guessing with the remaining 3 with probability 0.25

- or guessing from the 4 choices with probability 0.25

then defining the event R = getting the answer right, we have

P (R) = probability of knowing the answer*probability of getting the question right if knowing the answer + probability of eliminating one answer * probability of getting the question right if eliminates one answer + probability of guessing the 4 choices * probability of getting the question right if guessing the 4 choices

thus

P (R) = 0.5*1 + 0.25 * 1/3 + 0.25*1/4 = 0.6458

then we use conditional probability through the theorem of Bayes. Defining K = student knew the answer

then

P (K/R) = P (K∩R) / P (R) = 0.5*1/0.6458 = 0.7742 (77.42%)

where

P (K∩R) = probability that the student knew the answer and answers the question correctly

P (K/R) = probability that the student knew the answer given that he answered the question correctly