$12$ balls are distributed among $3$ boxes. The probability that the first box will contain exactly $3$ balls is:

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 ..........

Let $B_{i} (i=1, 2, 3)$ be three independent events in a sample space. The probability that only $B_{1}$ occurs is $\alpha$,only $B_{2}$ occurs is $\beta$,and only $B_{3}$ occurs is $\gamma$. Let $p$ be the probability that none of the events $B_{i}$ occurs,and these $4$ probabilities satisfy the equations $(\alpha - 2\beta)p = \alpha\beta$ and $(\beta - 3\gamma)p = 2\beta\gamma$ (All the probabilities are assumed to lie in the interval $(0, 1)$). Then $\frac{P(B_{1})}{P(B_{3})}$ is equal to ..........

Functions are formed from the set $A = \{a_1, a_2, a_3\}$ to another set $B = \{b_1, b_2, b_3, b_4, b_5\}$. If a function is selected at random,the probability that it is a one-one function is

Two squares are chosen at random on a chess-board. The probability that they have a side in common is

If $A$ and $B$ are independent events and $P(A) = p, P(B) = 2p$,and $P(\text{exactly one from } A \text{ and } B) = \frac{5}{9}$,then find the value of $p$.