$A$ pair of $12$-sided fair dice with faces numbered $1, 2, 3, \ldots, 12$ is rolled. The probability that the sum of the numbers appearing has a remainder of $2$ when divided by $9$ is

If two cards are drawn at random simultaneously from a pack of $52$ playing cards, then the probability of getting a face card and a spade card other than the face card 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

Let $a, b, c \in \{1, 2, 3, 4\}$. If the probability that $ax^2 + 2\sqrt{2}bx + c > 0$ for all $x \in R$ is $m/n$, where $gcd(m, n) = 1$, then $m + n$ is equal to . . . . . . .

$A$ bag contains $2$ white,$3$ green,and $5$ red balls. If three balls are drawn one after the other without replacement,then the probability that the last ball drawn was red is

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