$25 \%$ of the population are smokers. $A$ smoker has $27$ times more chances to develop lung cancer than a non-smoker. $A$ person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of $k$ is $.............$

In a bolt factory,machines $A, B$ and $C$ manufacture respectively $20 \%$,$30 \%$ and $50 \%$ of the total bolts. Of their output,$3 \%$,$4 \%$ and $2 \%$ are respectively defective bolts. $A$ bolt is drawn at random from the product. If the bolt drawn is found to be defective,then the probability that it is manufactured by the machine $C$ is

In an entrance test, there are multiple-choice questions. There are four possible answers to each question, of which one is correct. The probability that a student knows the answer to a question is $9/10$. If he gets the correct answer to a question, then the probability that he was guessing 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:

There are $10$ coins in a box, out of which $8$ are normal and the remaining are with heads on both sides. $A$ coin is chosen at random from the box and tossed $6$ times. If it shows heads each time, then the probability that the selected coin has heads on both sides is

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