$A$ fair die is rolled indefinitely. Player $A$ wins if two consecutive rolls show $3$ or $5$, and player $B$ wins if two consecutive rolls show $1$ or $2$ or $4$ or $6$. The probability that player $A$ wins is:

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:

$A$ and $B$ throw a pair of dice alternately and they note the sum of the numbers appearing on the dice. $A$ wins if he throws $6$ before $B$ throws $7$,and $B$ wins if he throws $7$ before $A$ throws $6$. If $A$ begins,the probability of $A$ winning is:

Cards are drawn one after the other without replacement from a well-shuffled pack of cards until an ace card appears. If the probability that exactly $5$ cards are drawn before the first ace card appears is $\frac{4}{49}\left(\frac{p_1 \cdot p_2 \cdot p_3}{p_4 \cdot p_5 \cdot p_6}\right)$, where $p_i$ is prime for $i=1, 2, 3, 4, 5, 6$, then $(\max \{p_i\} - \min \{p_i\}) = $

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 $IIT$?

Let $A, B, C$ be three pairwise independent events of a random experiment. If $P(\bar{B} \cup \bar{C}) = \frac{1}{2}$, $P(A) > 0$, $P(B) = b$, and $P(C) = c$, then $P((\bar{B} \cap \bar{C}) \mid A) = $