Events $E_{1}$ and $E_{2}$ form a partition of the sample space $S$. $A$ is any event such that $P(E_{1}) = P(E_{2}) = \frac{1}{2}$,$P(E_{2} | A) = \frac{1}{2}$,and $P(A | E_{2}) = \frac{2}{3}$. Then $P(E_{1} | A)$ is:

$A$ company produces $10,000$ items per day. On a particular day,$2500$ items were produced on machine $A$,$3500$ on machine $B$,and $4000$ on machine $C$. The probability that an item produced by machines $A, B, C$ is defective is $2 \%$,$3 \%$,and $5 \%$,respectively. If one item is selected at random from the output and is found to be defective,then the probability that it was produced by machine $C$ is:

There are two coins,one unbiased with probability $\frac{1}{2}$ of getting heads and the other one is biased with probability $\frac{3}{4}$ of getting heads. $A$ coin is selected at random and tossed. It shows heads up. Then,the probability that the unbiased coin was selected is

In a test,a student either guesses,copies,or knows the answer to a multiple-choice question with four choices. The probability that he guesses is $1/3$ and the probability that he copies the answer is $1/6$. The probability that his answer is correct,given that he copied it,is $1/8$. The probability that he knew the answer to the question,given that he answered it correctly,is

$A$ man is known to speak the truth $3$ out of $4$ times. He throws a die and reports that it is a six. The probability that it is actually a six,is

There are $3$ bags which are known to contain $2$ white and $3$ black balls; $4$ white and $1$ black balls and $3$ white and $7$ black balls respectively. $A$ ball is drawn at random from one of the bags and found to be a black ball. Then the probability that it was drawn from the bag containing the most black balls is