$A$ man is known to speak the truth $7$ out of $10$ times. After throwing a die with $100$ faces marked $1, 2, 3, \dots, 100$,the man reports that he got a prime number. What is the probability that it is actually a prime number?

$A$ bag contains $2n+1$ coins. It is known that $n$ of these coins have a head on both sides, whereas the remaining $n+1$ coins are fair. $A$ coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is $\frac{31}{42}$, then $n$ is equal to

In a certain college, $4 \%$ of men and $1 \%$ of women are taller than $1.8 \ m$. Also, $60 \%$ of students are women. If a student selected at random is found to be taller than $1.8 \ m$, then the probability that the student is a woman is: (in $/ 11$)

$A$ manufacturing company of bulbs has $3$ units $A, B$ and $C$ which produce $25 \%$,$35 \%$ and $40 \%$ of the bulbs respectively. Out of the bulbs produced by $A, B, C$ units,$5 \%, 4 \%$ and $2 \%$ are defective respectively. If a bulb is chosen at random and found to be defective,then the probability that it is produced by unit $B$ 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:

$A$ company has two plants $A$ and $B$ to manufacture motorcycles. $60 \%$ of motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. $80 \%$ of the motorcycles manufactured at plant $A$ are rated of standard quality,while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of standard quality. $A$ motorcycle picked up randomly from the total production is found to be of standard quality. If $p$ is the probability that it was manufactured at plant $B$,then $126 p$ is