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

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?

Let $S = \{w_1, w_2, \ldots\}$ be the sample space associated with a random experiment. Let $P(w_n) = \frac{P(w_{n-1})}{2}$ for $n \geq 2$. Let $A = \{2k + 3\ell : k, \ell \in \mathbb{N}\}$ and $B = \{w_n : n \in A\}$. Then $P(B)$ is equal to:

Let $X$ and $Y$ be two events such that $P(X \cup Y) = P(X \cap Y)$.
Statement $1$: $P(X \cap Y') = P(X' \cap Y) = 0$.
Statement $2$: $P(X) + P(Y) = 2P(X \cap Y)$.

If two cards are drawn at random simultaneously from a pack of $52$ playing cards, then the probability of getting a face card and a spade card other than the face card is:

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