Four persons $A, B, C$ and $D$ throw an unbiased die, turn by turn in succession till one gets an even number and wins the game. What is the probability that $A$ wins the game if $A$ begins?

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?

Five different books are to be distributed among four students randomly. The probability that each child gets at least one book is

Let $S = \{w_1, w_2, \ldots\}$ be the sample space associated with a random experiment. Let $P(w_n) = \frac{P(w_{n-1})}{2}$ for $n \geq 2$. Let $A = \{2k + 3\ell : k, \ell \in \mathbb{N}\}$ and $B = \{w_n : n \in A\}$. Then $P(B)$ is equal to:

$A$ and $B$ are independent events with $P(A)=\frac{1}{4}$ and $P(A \cup B)=2 P(B)-P(A)$,then $P(B)$ is

In a battery manufacturing factory, machines $P$, $Q$ and $R$ manufacture $20 \%$, $30 \%$ and $50 \%$ respectively of the total output. The chances that a defective battery is produced by these machines are $1 \%$, $1.5 \%$ and $2 \%$ respectively. If a battery is selected at random from the production, then the probability that it is defective is