$A$ letter is known to have arrived by post either from $KANPUR$ or from $ANANTPUR$. On the envelope,just two consecutive letters $AN$ are visible. The probability that the letter came from $ANANTPUR$ is:

$A$ man speaks truth $2$ out of $3$ times. He picks one of the natural numbers in the set $S=\{1, 2, 3, 4, 5, 6, 7\}$ and reports that it is even. The probability that it is actually even 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:

$A$ man is known to speak the truth $2$ out of $3$ times. If he throws a die and reports that it is $6$,then the probability that it is actually $5$ is:

An item is tested on a device for its defectiveness. The probability that such an item is defective is $0.3$. The device gives an accurate result in $8$ out of $10$ such tests. If the device reports that an item tested is not defective,then the probability that it is actually defective is:

$A$ student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question,he gives the correct answer. Assume that the probability of the student giving the correct answer for a question,given that he has guessed it,is $\frac{1}{2}$. Also assume that the probability of the answer for a question being guessed,given that the student's answer is correct,is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question is