If $A$ and $B$ are events of a sample space such that $P(A \cup B) = \frac{3}{4}$, $P(A \cap B) = \frac{1}{4}$, and $P(\bar{A}) = \frac{2}{3}$, then $P(\bar{A} \cap B)$ is

The probabilities of $A, B,$ and $C$ solving a problem are $\frac{1}{3}, \frac{2}{7},$ and $\frac{3}{8}$ respectively. If all three try to solve the problem simultaneously,the probability that exactly one of them will solve it is:

$A$ and $B$ are two events such that $P(A)=0.54$,$P(B)=0.69$ and $P(A \cap B)=0.35$. Find $P(A \cap B^{\prime})$.

$A$ box contains $4$ white pens and $2$ black pens. Another box contains $3$ white pens and $5$ black pens. If one pen is selected from each box,then the probability that both the pens are white is equal to

$A$ card is selected from a pack of $52$ cards. How many points are there in the sample space?

Two dice are thrown. The events $A, B$ and $C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\leq 5$.
State true or false: (give reason for your answer)
Statement: $A', B', C$ are mutually exclusive and exhaustive.