Probability — NDA Mathematics PYQs

Consider the following for the two (02) items that follow :
Let $X$ be a random variable following binomial distribution with parameters $n=6$ and $p=k$. Further, $9P(X=4)=P(X=2)$.

What is the value of $k$?

Consider the following for the two (02) items that follow :
Let $X$ be a random variable following binomial distribution with parameters $n=6$ and $p=k$. Further, $9P(X=4)=P(X=2)$.

What is the value of $P(X=3)$?

Consider the following for the two (02) items that follow :
A committee of $6$ members is formed from a group of $7$ gentlemen and $4$ ladies.

What is the probability that the committee includes exactly $3$ gentlemen?

Consider the following for the two (02) items that follow :
A committee of $6$ members is formed from a group of $7$ gentlemen and $4$ ladies.

What is the probability that the committee includes at least $2$ ladies?

Consider the following for the two (02) items that follow :
The probabilities that $A$, $B$ and $C$ become managers are $\frac{3}{10}$, $\frac{1}{2}$ and $\frac{4}{5}$ respectively. The probabilities that bonus scheme will be introduced if $A$, $B$ and $C$ become managers are $\frac{4}{9}$, $\frac{2}{9}$ and $\frac{1}{3}$ respectively.

What is the probability that the bonus scheme will be introduced?

Consider the following for the two (02) items that follow :
The probabilities that $A$, $B$ and $C$ become managers are $\frac{3}{10}$, $\frac{1}{2}$ and $\frac{4}{5}$ respectively. The probabilities that bonus scheme will be introduced if $A$, $B$ and $C$ become managers are $\frac{4}{9}$, $\frac{2}{9}$ and $\frac{1}{3}$ respectively.

If the bonus scheme has been introduced, then what is the probability that the manager appointed was $B$?

If $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{2}$ and $P(A\cap B)=\frac{1}{4}$, then what is $P(B|A^c)$ equal to?

If $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{2}$ and $P(A\cap B)=\frac{1}{4}$, then what is $P(A^c\cap B^c)$ equal to?

If two fair dice are tossed, then what is the probability that the sum of the numbers on the faces of the dice is strictly greater than $7$?

The probability of a man hitting a target is $\frac{1}{5}$. If the man fires $7$ times, then what is the probability that he hits the target at least twice?