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\}) = $

Let $X$ and $Y$ be two events such that $P(X \cup Y) = P(X \cap Y)$.
Statement $1$: $P(X \cap Y') = P(X' \cap Y) = 0$.
Statement $2$: $P(X) + P(Y) = 2P(X \cap Y)$.

$A$ bag contains $7$ different black balls and $10$ different red balls. If balls are drawn one by one randomly until all black balls are drawn,what is the probability that this process is completed in the $12^{th}$ draw?

If $A$ and $B$ are two events such that $P(A \cup B) \geq \frac{3}{4}$ and $\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}$, then which of the following is true?

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 three bags $B_1$,$B_2$,and $B_3$ containing $2$ Red and $3$ White,$5$ Red and $5$ White,and $3$ Red and $2$ White balls respectively. $A$ ball is drawn from bag $B_1$ and placed in bag $B_2$,then a ball is drawn from bag $B_2$ and placed in bag $B_3$,then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed,if the same colour balls are used in the first and second transfers (assume all balls to be distinct),is