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 ..........

If $A$ and $B$ are two events such that $P(A) = \frac{1}{2}$ and $P(B) = \frac{2}{3},$ then

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?

In a random experiment,a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to

In a game, two dice are thrown simultaneously by a person $A$ and two cards are drawn at random simultaneously from a pack of $52$ playing cards by a person $B$. They win the game if $A$ gets a prime score as the sum of the numbers appearing on both the dice and $B$ gets a face card and a card having a prime number. Then the probability that both $A$ and $B$ win is:

If the integers $m$ and $n$ are chosen at random between $1$ and $100$,then the probability that a number of the form $7^m + 7^n$ is divisible by $5$ equals