If $P(B) = \frac{3}{4}$,$P(A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P(\bar{A} \cap B \cap \bar{C}) = \frac{1}{3}$,then $P(B \cap C)$ is

$A$ die is rolled three times. The probability of getting their sum equal to a prime number of the form $4n+1$ is

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

You are given a box containing $20$ cards. Out of these,$10$ cards have the letter $I$ printed on them,and the other $10$ cards have the letter $T$ printed on them. If you draw three cards one after another with replacement,what is the probability of forming the word $IIT$?

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:

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