An unbiased coin is tossed. If the result is a head,a pair of unbiased dice is rolled and the sum of the numbers on the two faces is noted. If the result is a tail,a card from a well-shuffled pack of eleven cards numbered $2, 3, 4, \dots, 12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is:

Two persons $A$ and $B$ take part in a shooting competition. $A$ can hit the target with a probability of $0.6$. $B$ can hit the target with a probability of $0.8$. $A$ takes the first shot,after which they shoot alternately. The probability that $A$ wins the competition is

If $10$ different balls are to be placed in $4$ distinct boxes at random,then the probability that two of these boxes contain exactly $2$ and $3$ balls is

Box $1$ contains three cards bearing numbers $1, 2, 3$; box $2$ contains five cards bearing numbers $1, 2, 3, 4, 5$; and box $3$ contains seven cards bearing numbers $1, 2, 3, 4, 5, 6, 7$. $A$ card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{\text{th}}$ box,$i = 1, 2, 3$.
$1.$ The probability that $x_1 + x_2 + x_3$ is odd is:
$(A) \frac{29}{105}$ $(B) \frac{53}{105}$ $(C) \frac{57}{105}$ $(D) \frac{1}{2}$
$2.$ The probability that $x_1, x_2, x_3$ are in an arithmetic progression is:
$(A) \frac{9}{105}$ $(B) \frac{10}{105}$ $(C) \frac{11}{105}$ $(D) \frac{7}{105}$
Give the answers for question $1$ and $2$.

$A$ box $B_1$ contains $1$ white ball,$3$ red balls and $2$ black balls. Another box $B_2$ contains $2$ white balls,$3$ red balls and $4$ black balls. $A$ third box $B_3$ contains $3$ white balls,$4$ red balls and $5$ black balls.
$1.$ If $1$ ball is drawn from each of the boxes $B_1, B_2$ and $B_3$,the probability that all $3$ drawn balls are of the same colour is
$(A)$ $\frac{82}{648}$ $(B)$ $\frac{90}{648}$ $(C)$ $\frac{558}{648}$ $(D)$ $\frac{566}{648}$
$2.$ If $2$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red,the probability that these $2$ balls are drawn from box $B_2$ is
$(A)$ $\frac{116}{181}$ $(B)$ $\frac{126}{181}$ $(C)$ $\frac{65}{181}$ $(D)$ $\frac{55}{181}$
Choose the correct options for question $1$ and $2$.

$A$ and $B$ are two independent events of a random experiment and $P(A) > P(B)$. If the probability that both $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$,then the probability of the occurrence of $B$ is