Question

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4 . What is the probability that the student knows the answer given that he answered it correctly?

Answer 1

Let E1 : the event that the student knows the answer and E2 : the event that the student guesses the answer.

Then, E1 and E2 are mutually exclusive and exhaustive. Moreover,

P(E1) = 3/4 and P(E2) = 1/4

Let E : the answer is correct. The probability that the student answered correctly, given that he knows the answer, is 1 i.e., P P(E/E1) = 1

Probability that the students answered correctly, given that the he guessed, is 1/4

i.e., P(E/E2) = 1/4

By using Baye’* theorem, we obtain

P(E1/E) = (P(E1)P(E/E1))/(P(E1)P(E/E1) + P(E2)P(E/E2))

= (3/4 x 1)/(3/4 x 1 + 1/4 x 1/4) = (3/4)/(3/4 + 1/16) = (12/16)/(12 + 1)/16 = 12/13