For independent events $A$ and $B$,if $P(A) = \frac{1}{2}$ and $P(A \cup B) = \frac{3}{5}$,then $P(B) =$ . . . . . . .

$A$ bag contains $19$ red balls and $19$ black balls. Two balls are chosen at a time repeatedly and discarded if they are of the same colour,but if they are different,the black ball is discarded and the red ball is returned to the bag. The probability that this process will terminate with one red ball is

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:

Three six-faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is $k$ $(3 \le k \le 8)$ is:

There are four fair dice $D_1, D_2, D_3$ and $D_4$. Each has six faces numbered $1, 2, 3, 4, 5$ and $6$. They are rolled one by one. What is the probability that the number shown on $D_4$ is equal to at least one of the numbers shown on $D_1, D_2$ and $D_3$ (in $/216$)?

If $a$ and $b$ are chosen randomly from the set $\{1, 2, 3, 4, 5, 6\}$ with replacement,then the probability that $\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x}}}{2}} \right)^{\frac{2}{x}}}=6$ is