The probability of solving a problem by three persons $A, B$ and $C$ independently is $\frac{1}{2}, \frac{1}{4}$ and $\frac{1}{3}$ respectively. Then the probability that the problem is solved by exactly two of them is

Each of $a$ and $b$ can take values $1$ or $2$ with equal probability. The probability that the equation $ax^{2} + bx + 1 = 0$ has real roots is equal to

Two dice are tossed. The probability that the total score is a prime number is

$A$ special lottery is to be held to select a student who will live in the only deluxe room available in a hostel. $100$ $III$ year,$150$ $II$ year,and $200$ $I$ year students have applied for the room. Each $III$ year student's name is placed in the lottery $3$ times,each $II$ year student's name $2$ times,and each $I$ year student's name $1$ time. The probability that a $III$ year student gets the room is:

If $A$ and $B$ are events of a random experiment such that $P(A \cup B) = \frac{4}{5}$,$P(\bar{A} \cup \bar{B}) = \frac{7}{10}$,and $P(B) = \frac{2}{5}$,then $P(A)$ equals

The probability that Ajay will not appear in the $JEE$ exam is $p = \frac{2}{7}$,while the probability that both Ajay and Vijay will appear in the exam is $q = \frac{1}{5}$. Then the probability that Ajay will appear in the exam and Vijay will not appear is: