Suppose four balls labelled $1, 2, 3, 4$ are randomly placed in boxes $B_1, B_2, B_3, B_4$. The probability that exactly one box is empty is

Two persons $A$ and $B$ take turns in throwing a pair of dice. The first person to throw a sum of $9$ with both dice will win the game. If $A$ throws first,then the probability that $B$ wins the game is:

$A$ basket contains $5$ apples and $7$ oranges and another basket contains $4$ apples and $8$ oranges. One fruit is picked out from each basket. Find the probability that the fruits are both apples or both oranges.

$A$ and $B$ are independent events. The probability that both $A$ and $B$ occur is $\frac{1}{20}$ and the probability that neither of them occurs is $\frac{3}{5}$. The probability of occurrence of $A$ is

Consider the system of equations $ax+by=0, cx+dy=0$,where $a, b, c, d \in \{0, 1\}$.
$STATEMENT-1$: The probability that the system of equations has a unique solution is $3/8$.
$STATEMENT-2$: The probability that the system of equations has a solution is $1$.

Let $X$ and $Y$ be events such that $P(X \cup Y) = P(X \cap Y).$
Statement-$1$: $P(X \cap Y) = P(X' \cap Y') = 0$
Statement-$2$: $P(X) + P(Y) = 2P(X \cap Y).$