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 he passes either in tests $I$ and $II$ or tests $I$ and $III$. The probabilities of the student passing in tests $I, II, III$ are $p, q$ and $\frac{1}{2}$ respectively. If the probability that the student is successful is $\frac{1}{2}$,then

If $P(A)=0.59, P(B)=0.30$ and $P(A \cap B)=0.21$,then $P(A^{\prime} \cap B^{\prime})$ is equal to

Let $A$ and $B$ be two events such that $P(\overline{A \cup B}) = \frac{1}{6}$,$P(A \cap B) = \frac{1}{4}$ and $P(\bar{A}) = \frac{1}{4}$,where $\bar{A}$ stands for the complement of the event $A$. Then the events $A$ and $B$ are

If two dice are thrown simultaneously,then the probability that the sum of the numbers which come up on the dice is more than $5$ is:

From a well-shuffled pack of cards,one card is drawn at random. The probability that the card drawn is an ace is