Two events $A$ and $B$ will be independent,if

Three dice are thrown simultaneously and the sum of the numbers appeared on them is noted. If $A$ is the event of getting a sum greater than $14$ and $B$ is the event of getting a sum which is a multiple of $3$,then $P(A \cap \overline{B}) + P(\overline{A} \cap B) = $

Let $A$ denote the set of all $4$-digit natural numbers with no digit being $0$. Let $B \subset A$ consist of all numbers $x$ such that no permutation of the digits of $x$ gives a number that is divisible by $4$. Then the probability of drawing a number from $B$ with all even digits is

Let $p$ denote the probability that a man aged $x$ years will die in a year. The probability that out of $n$ men $A_1, A_2, A_3, ..., A_n$ each aged $x$,$A_1$ will die in a year and will be the first to die,is

$S$ is the sample space and $A, B$ are two events of a random experiment. Match the items of List-$A$ with the items of List-$B$.

List-$A$	List-$B$
$(I)$ $A, B$ are mutually exclusive events	$(i)$ $P(A \cap B) = P(B) - P(\bar{A})$
$(II)$ $A, B$ are independent events	$(ii)$ $P(A) \leq P(B)$
$(III)$ $A \cap B = A$	$(iii)$ $P(\frac{\bar{A}}{B}) = 1 - P(A)$
$(IV)$ $A \cup B = S$	$(iv)$ $P(A \cup B) = P(A) + P(B)$
	$(v)$ $P(A) + P(B) = 2$

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 . . . . . . .