Two dice are thrown simultaneously. What is the probability of obtaining a sum of the numbers less than $11$?

$2$ boys and $2$ girls are in Room $X$,and $1$ boy and $3$ girls are in Room $Y$. Specify the sample space for the experiment in which a room is selected and then a person.

$A$ problem is given to $3$ students $A, B$ and $C$ whose chances of solving it are $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$ respectively. Then,the probability of the problem being solved by exactly one of them,if all the three try independently,is

If $A$ and $B$ are any two events such that $P(\overline{A \cup B}) = \frac{1}{6}$,$P(A \cap B) = \frac{1}{4}$,and $P(\bar{A}) = \frac{1}{4}$,then the events $A$ and $B$ are

If two balanced dice are tossed once,the probability of the event that the sum of the integers appearing on the upper faces of the two dice is $9$ is:

If two dice are thrown simultaneously,then the probability that $1$ comes on the first die is