$A$ bag $P$ contains $4$ red and $5$ black balls,another bag $Q$ contains $3$ red and $6$ black balls. If one ball is drawn at random from bag $P$ and two balls are drawn from bag $Q$,then the probability that out of the three balls drawn two are black and one is red,is

$A$ 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 $2$.

$A$ boy throws an unbiased die. Whenever he gets $1$ on the die, he has a further chance to throw it once again immediately. The probability that the boy gets a score of $7$ in this process is

$A$ and $B$ toss a fair coin each simultaneously $50$ times. The probability that both of them will not get tail at the same toss is

For the three events $A, B$ and $C$,$P$ (exactly one of the events $A$ or $B$ occurs) = $P$ (exactly one of the events $B$ or $C$ occurs) = $P$ (exactly one of the events $C$ or $A$ occurs) = $p$ and $P$ (all the three events occur simultaneously) = $p^2$,where $0 < p < 1/2$. Then the probability of at least one of the three events $A, B$ and $C$ occurring is

Fifteen football players of a club-team are given $15$ $T$-shirts with their names written on the backside. If the players pick up the $T$-shirts randomly,then the probability that at least $3$ players pick the correct $T$-shirt is