For any two events $A$ and $B$ in a sample space,which of the following is always true?

$A$ 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:

In a game,a person wins $5$ rupees for getting a number greater than $4$ and loses $1$ rupee otherwise,when a fair die is thrown. $A$ man participates in the game and decides to quit as soon as he gets a number greater than $4$. If he plays for a maximum of $3$ throws,what is the expected value (mean value) of the amount he wins or loses?

$A$ coin is tossed $8$ times. If the probability that exactly $4$ heads appear in the first six tosses and exactly $3$ heads appear in the last five tosses is $p$, then $96p$ is equal to ————

Let $A, B,$ and $C$ be $3$ independent events such that $P(A) = 1/3, P(B) = 1/2,$ and $P(C) = 1/4$. Find the probability that exactly $2$ of the $3$ events occur.

There are $2$ shelves. One shelf has $5$ Physics and $3$ Biology books,and the other has $4$ Physics and $2$ Biology books. What is the probability of drawing a Physics book?