Two persons $A$ and $B$ take turns in throwing a pair of dice. The first person to throw a sum of $9$ with both dice will win the game. If $A$ throws first,then the probability that $B$ wins the game is:

Consider a system of equations $ax + by = 0$ and $cx + dy = 0$,where $a, b, c, d \in \{0, 1\}$.
Statement $-1$: The probability that the system of equations has a solution is $1$.
Statement $-2$: The probability that the system of equations has a unique solution is $\frac{3}{8}$.

$A$ basket contains $5$ apples and $7$ oranges and another basket contains $4$ apples and $8$ oranges. One fruit is picked out from each basket. Find the probability that the fruits are both apples or both oranges.

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 . . . . . . .

There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order until both the faulty machines are identified. The probability that only two tests are needed is:

Let $A$ and $B$ be two events such that $P(A \cap B) = \frac{1}{6}$,$P(A \cup B) = \frac{31}{45}$,and $P(\bar{B}) = \frac{7}{10}$. Then which of the following is true?