Question

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

Answer 1

Let, P( A ), P( B ) and P( C ) be the probabilities that are defined below,

The probability of student answer is P( A ).

The probability of student guesses is P( B ).

The probability of student answers correctly is P( C ).

The probability that the student knows the answer, if he answered it correctly that is P( A C ),

P( A C )= P( A )P( C A ) P( B )⋅P( C B )+P( A )⋅P( C A ) (1)

The probability that student knows the answer is,

P( A )= 3 4

The probability that the student answer correctly, given that he knows the answer,

P( C A )=1

The probability that student guess the answer is,

P( B )= 1 4

The probability that the student answer correctly, given that he guesses,

P( C B )= 1 4

Put these values in equation (1),

P( A C )= ( 3 4 ×1 ) ( 1 4 × 1 4 )+( 3 4 ×1 ) = 3 4 1 16 + 3 4 = 3 4 13 16 = 12 13

Thus, the required probability is 12 13 .