$A$ bag contains $2n+1$ coins. It is known that $n$ of these coins have heads on both sides,whereas the other $n+1$ coins are fair. One coin is selected at random and tossed. If the probability that the toss results in heads is $\frac{31}{42}$,then the value of $n$ is

In an entrance test, there are multiple-choice questions. Each question has four possible answers, only one of which is correct. The probability that a student knows the answer to a question is $90\%$. If the student gets the correct answer to a question, what is the probability that they were guessing?

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:

$A$ bag contains $10$ balls out of which $k$ are red and $(10-k)$ are black, where $0 \le k \le 10$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains $1$ red and $9$ black balls is:

$A$ person is known to speak the truth $3$ out of $4$ times. If that person picks a card at random from a pack of $52$ cards and reports that it is a king,then the probability that it is actually a king is

$A$ person goes to office by car, scooter, bus, and train, the probabilities of which are $1/7, 3/7, 2/7,$ and $1/7$ respectively. The probability that he reaches office late if he takes a car, scooter, bus, or train is $2/9, 1/9, 4/9,$ and $1/9$ respectively. Given that he reached the office in time, the probability that he travelled by car is: (in $/7$)