$A$ number $n$ is randomly selected from the set $\{1, 2, 3, \dots, 1000\}$. The probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer is:

$A$ boy throws an unbiased die. Whenever he gets $1$ on the die, he has a further chance to throw it once again immediately. The probability that the boy gets a score of $7$ in this process is

There are $3$ bags $A, B$ & $C$. Bag $A$ contains $1$ Red & $2$ Green balls,bag $B$ contains $2$ Red & $1$ Green balls and bag $C$ contains only $1$ Green ball. One ball is drawn from bag $A$ & put into bag $B$,then one ball is drawn from $B$ & put into bag $C$,& finally one ball is drawn from bag $C$ & put into bag $A$. When this operation is completed,what is the probability that bag $A$ contains $2$ Red & $1$ Green balls?

In a throw of a dice,the probability of getting a $1$ in an even number of throws is:

If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B) = \frac{1}{6}$ and $P(\bar{A} \cap \bar{B}) = \frac{1}{3}$,then $P(A)$ is equal to (Here,$\bar{E}$ is the complement of the event $E$)

Let $a, b$ and $c$ denote the outcomes of three independent rolls of a fair tetrahedral die,whose four faces are marked $1, 2, 3, 4$. If the probability that the quadratic equation $ax^2 + bx + c = 0$ has real roots is $\frac{m}{n}$,where $\operatorname{gcd}(m, n) = 1$,then $m + n$ is equal to ..........