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?

If $n$ numbers are chosen at random from the set $\{1, 2, 3, \dots, 1000\}$,what is the probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer?

$A$ pair of $12$-sided fair dice with faces numbered $1, 2, 3, \ldots, 12$ is rolled. The probability that the sum of the numbers appearing has a remainder of $2$ when divided by $9$ is

An urn $A$ contains $4$ white and $1$ black ball; urn $B$ contains $3$ white and $2$ black balls and urn $C$ contains $2$ white and $3$ black balls. One ball is transferred randomly from $A$ to $B$; later one ball is transferred randomly from $B$ to $C$. Finally,if a ball is drawn randomly from $C$,then the probability that it is a black ball is

$A$ coin is tossed $2020$ times. The probability of getting head on $1947^{\text{th}}$ toss is

Let a computer program generate only the digits $0$ and $1$ to form a string of binary numbers. The probability of occurrence of $0$ at even places is $\frac{1}{2}$ and the probability of occurrence of $0$ at odd places is $\frac{1}{3}$. Then the probability that $'10'$ is followed by $'01'$ is equal to: