$A$ locker can be opened by dialing a fixed three-digit code (between $000$ and $999$). $A$ stranger who does not know the code tries to open the locker by dialing three digits at random. The probability that the stranger succeeds at the $k^{th}$ trial is

For any two events $A$ and $B$ in a sample space,which of the following is always true?

Let there be three independent events $E_{1}, E_{2}$ and $E_{3}$. The probability that only $E_{1}$ occurs is $\alpha$,only $E_{2}$ occurs is $\beta$ and only $E_{3}$ occurs is $\gamma$. Let $p$ denote the probability that none of the events occur,which satisfies the equations $(\alpha - 2\beta)p = \alpha\beta$ and $(\beta - 3\gamma)p = 2\beta\gamma$. All the given probabilities are assumed to lie in the interval $(0, 1)$. Then,$\frac{\text{Probability of occurrence of } E_{1}}{\text{Probability of occurrence of } E_{3}}$ is equal to ..........

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 $|X|$ denote the number of elements in set $X$. Let $S = \{1, 2, 3, 4, 5, 6\}$ be a sample space,where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$,then the number of ordered pairs $(A, B)$ such that $1 \leq |B| < |A|$ equals:

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?