Question

A student answers a multiple choice examination with two questions that have four possible answers each. Suppose that the probability that the student knows the answer to a question is 0.80 and the probability that the student guesses is 0.20. If the student guesses, the probability of guessing the correct answer is 0.25. The questions are independent, that is, knowing the answer on one question is not influenced by the other question.

(a) What is the probability that the both questions will be answered correctly?

(b) If answered correctly, what is the probability that the student really knew the correct answer to both questions?

Answer 1

P(   know answer   ) =    0.8
P(   guess   ) =    0.2

          
P(   correct answer   | know answer)=   1
P(   correct answer   | guess)=   0.25

a)

P(correct answer) = P(know answer) * P(correct answer| know answer) + P(guess) *P(correct answer| guess) =                                    0.8*1+0.2*0.25=               0.85

P(both answers correct) = 0.85*0.85 = 0.7225

b)

P(know answer| correct answer) = P(know answer)*P(correct answer| know answer)/P(correct answer)=               0.8*1/0.85=       0.9412

probability that the student really knew the correct answer to both questions = 0.9412^2 =0.8858