Question

In answering a question on a multiple choice test a student either knows the answeror guesses.Let $\frac{3}{4}$ be the probability that he knows the answer and $\frac{1}{4}$ be the probability that he guesses.Assuming that the student who guessws the answer,will be correct with probability $\frac{1}{4}$.What is the probability that a student knows the answer given that he answered it correctly?

Answer 1

The correct answer is: $\frac{12}{13}$
Let $E_1$=the examiee knows the answer, $E_2$=the examinee guesses the answer and A=the examinee answers correctly
Now $P(E_1)=\frac{3}{4},P(E_2)=\frac{1}{4},$
Since $E_1$ and $E_2$ are mutually exclusive events and exhaustive events, and if $E_2$ has already occurred, then the examinee guesses,
Therefore the probability that he answers correctly given that he has made a guess is $\frac{1}{4}$ i.e., $P(A|E_2)=\frac{1}{4}$
And $P(A|E_2)$=P(answers correctly given that he knew the answer)=1
Therefore,by Bayes' theorem,
$P(E_1|A)=\frac{P(E_1)P(A|E_1)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)}$
$=\frac{\frac{3}{4}.1}{\frac{3}{4}.1+\frac{1}{4}.\frac{1}{4}}$
$=\frac{\frac{3}{4}}{\frac{3}{4}+\frac{1}{16}}$
$=\frac{\frac{3}{4}}{\frac{13}{16}}$
$=\frac{12}{13}$