Two persons $A$ and $B$ throw a fair die (six-faced cube with faces numbered from $1$ to $6$) alternately,starting with $A$. The first person to get an outcome different from the previous one thrown by the opponent wins. The probability that $B$ wins is:

There are $3$ bags $A, B$ & $C$. Bag $A$ contains $1$ Red & $2$ Green balls,bag $B$ contains $2$ Red & $1$ Green balls and bag $C$ contains only $1$ Green ball. One ball is drawn from bag $A$ & put into bag $B$,then one ball is drawn from $B$ & put into bag $C$,& finally one ball is drawn from bag $C$ & put into bag $A$. When this operation is completed,what is the probability that bag $A$ contains $2$ Red & $1$ Green balls?

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

Let $S = \{1, 2, \dots, 20\}$. $A$ subset $B$ of $S$ is said to be "nice" if the sum of the elements of $B$ is $203$. Then the probability that a randomly chosen subset of $S$ is "nice" is

For three events $A, B$ and $C$,$P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs}) = P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \frac{1}{4}$ and $P(\text{All the three events occur simultaneously}) = \frac{1}{16}$. Then the probability that at least one of the events occurs is:

$A, B, C$ are mutually exclusive events such that $P(A) = \frac{3x+1}{3}$,$P(B) = \frac{1-x}{4}$,and $P(C) = \frac{1-2x}{2}$. Then the set of possible values of $x$ is: