If $12$ identical balls are to be placed randomly in $3$ identical boxes,then the probability that one of the boxes contains exactly $3$ balls is

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:

First bag contains $3$ red and $5$ black balls and second bag contains $6$ red and $4$ black balls. $A$ ball is drawn from each bag. The probability that one ball is red and the other is black,is

Box-$I$ contains $3$ cards bearing numbers $1, 2, 3$; Box-$II$ contains $5$ cards bearing numbers $1, 2, 3, 4, 5$ and Box-$III$ contains $7$ cards bearing numbers $1, 2, 3, 4, 5, 6, 7$. One card is drawn at random from each of the boxes. If $x_i$ is the number on the card drawn from the $i^{\text{th}}$ box,$i=1, 2, 3$,then the probability that $x_1+x_2+x_3$ is odd is equal to

In a random experiment,a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to

In a game,two players $A$ and $B$ take turns throwing a pair of fair dice,starting with player $A$. The total score on the two dice in each throw is noted. $A$ wins the game if he throws a total of $6$ before $B$ throws a total of $7$,and $B$ wins the game if he throws a total of $7$ before $A$ throws a total of $6$. The game stops as soon as either of the players wins. The probability of $A$ winning the game is: