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

If $l, m$ represent any two elements (identical or different) of the set $\{1, 2, 3, 4, 5, 6, 7\}$,then the probability that $lx^2 + mx + 1 > 0$ for all $x \in R$ is

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

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$,then which of the following is true?
$(A)$ $P(E)=\frac{4}{5}, P(F)=\frac{3}{5}$
$(B)$ $P(E)=\frac{1}{5}, P(F)=\frac{2}{5}$
$(C)$ $P(E)=\frac{2}{5}, P(F)=\frac{1}{5}$
$(D)$ $P(E)=\frac{3}{5}, P(F)=\frac{4}{5}$

Three numbers are chosen at random,one after another with replacement,from the set $S = \{1, 2, 3, \ldots, 100\}$. Let $p_1$ be the probability that the maximum of the chosen numbers is at least $81$ and $p_2$ be the probability that the minimum of the chosen numbers is at most $40$.
$(1)$ The value of $\frac{625}{4} p_1$ is
$(2)$ The value of $\frac{125}{4} p_2$ is

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?