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 with probability 1/4. what is the probability that the student knows the answer given that he answers correctly.?

Explanation

Let `E_1 and E_2` be the respective events that the student knows the answer and he guesses the answer.
Let `A` be the event that the answer is correct.
so,`P(E_1)=3/4`
`P(E_2)=1/4`
The probability that the student answered correctly, given that he knows the answer, is `1`.
so, `P(A/E_1)=1`
Probability that the student answered correctly, given that he guessed, is `1/4`
so, `P(A/E_1)=1/4`
The probability that the student knows the answer, given that he answered it correctly, is given by,
Required probability `=P(E_1/A)=(P(E_1)P(A/E_1))/(P(E_1)P(A/E_1)+P(E_2)P(A/E_2))`
`=(3/4)/(3/4+1/4*1/4)`
`=(3/4)/(13/16)`
`=12/13`
`=12/13`
`=0.92`