Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^{N} < N!$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $4m - 3n$ is equal to $......$.

$A$ student appears for tests $I$, $II$, and $III$. The student is successful if they pass in tests ($I$ and $II$) $OR$ ($I$ and $III$). The probabilities of the student passing in tests $I$, $II$, and $III$ are respectively $p$, $q$, and $1/2$. If the probability of the student being successful is $1/2$, then:

$A$ number $n$ is chosen at random from $\{1, 2, 3, 4, \ldots, 1000\}$. The probability that $n$ is a number that leaves remainder $1$ when divided by $7$ is:

When three coins are tossed simultaneously,what is the probability of getting at least one head?

Let $A = \{1, 3, 5, 7, 9\}$ and $B = \{2, 4, 6, 8\}$. If an ordered pair $(a, b)$ is chosen at random from the Cartesian product $A \times B$,what is the probability that $a + b = 9$?

Three coins are tossed. Describe two events which are mutually exclusive.