Probability that a student knows the answer The Probality that a student knows the correct answer to a multiple choice question is 2/3 . If the student does not know the  answer , then the student guesses the answer . The probality of the guessed answer being correct is 1/4 . Given that the student has answered the questions correctly , the conditional probability that the student knows the correct answer is _
I started solving it . But I'm stuck .
Let 
a = student knows the  correct answer
b = guessed answer is correct
P(a)=2/3  P(a')=1/3
P(b)=1/4  P(b')=3/4
Let c = answered all questions correctly 
I guess 
P(c)=P(a)+P(a').P(b)
P(c)=(2/3) +((1/3).(1/4))
P(c) = 3/4
TO find 
P(a/c)=P(a n c)/p(c)
I dont know how to find p(a n c) . Please elaborate and Help !!!
 A: Let's define the events as:
\begin{eqnarray*}
A &=& \mbox{Student knows the correct answer} \\
C &=& \mbox{Student answered correctly.} \\
\end{eqnarray*}
You are asked to find a conditional probability, $P(A \mid C)$. We can use Bayes' Theorem to calculate it:
\begin{eqnarray*}
P(A \mid C) &=& \dfrac{P(C \mid A)P(A)}{P(C \mid A)P(A) + P(C \mid A^c)P(A^c)} \\
&& \\
&=& \dfrac{1 \times \frac{2}{3}}{1 \times \frac{2}{3} + \frac{1}{4} \times \frac{1}{3}} \\
&& \\
&=& \dfrac{8}{9}
\end{eqnarray*}
Note:


*

*$P(C \mid A) = 1$ because, given that the student knows the answer, the answer must be correct.

*Event $A^c$ is "the student guessed" because he guesses if he doesn't know the answer.
A: p(a n c) will be the probability that he knows the correct answer and has answered correctly which should be (2/3 * 1)
A: 2/3 of the questions he will know and answer correctly.
Of the remaining 1/3, the student will guess 1/4 of them correctly.
So, 2/3 + (1/4 x 1/3) = 3/4 of questions answered correctly
1/4 x 1/3 = 1/12, which is the percentage of questions guessed correctly.
What percentage of the total 3/4 is that 1/12?
To find out, just divide: 
So, the probability that he guessed: (1/12)/(3/4) = 1/9
and the probability that he knows the answer is: (11/12)/(3/4) = 8/9
or you could just do 1 - 1/9 = 8/9
