Five different books are to be distributed among four students randomly. The probability that each child gets at least one book 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 $a, b, c \in \{1, 2, 3, 4\}$. If the probability that $ax^2 + 2\sqrt{2}bx + c > 0$ for all $x \in R$ is $m/n$, where $gcd(m, n) = 1$, then $m + n$ is equal to . . . . . . .

$A$ random variable $X$ has the following probability distribution:
| $X=x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $P(X=x)$ | $0.15$ | $0.23$ | $0.12$ | $0.20$ | $0.08$ | $0.10$ | $0.05$ | $0.07$ |
For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 5\}$,find $P(E \cup F)$.

Box-$I$ contains $3$ cards bearing numbers $1, 2, 3$; Box-$II$ contains $5$ cards bearing numbers $1, 2, 3, 4, 5$ and Box-$III$ contains $7$ cards bearing numbers $1, 2, 3, 4, 5, 6, 7$. One card is drawn at random from each of the boxes. If $x_i$ is the number on the card drawn from the $i^{\text{th}}$ box,$i=1, 2, 3$,then the probability that $x_1+x_2+x_3$ is odd is equal to

If $P(B) = \frac{3}{4}$,$P(A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P(\bar{A} \cap B \cap \bar{C}) = \frac{1}{3}$,then $P(B \cap C)$ is