Let $\mathrm{X}$ and $\mathrm{Y}$ be two events such that $\mathrm{P}(\mathrm{X})=\frac{1}{3}, \mathrm{P}(\mathrm{X} \mid \mathrm{Y})=\frac{1}{2}$ and $\mathrm{P}(\mathrm{Y} \mid \mathrm{X})=\frac{2}{5}$. Then

$[A]$ $\mathrm{P}\left(\mathrm{X}^{\prime} \mid \mathrm{Y}\right)=\frac{1}{2}$   $[B]$ $\mathrm{P}(\mathrm{X} \cap \mathrm{Y})=\frac{1}{5}$    $[C]$ $\mathrm{P}(\mathrm{X} \cup \mathrm{Y})=\frac{2}{5}$    $[D]$ $\mathrm{P}(\mathrm{Y})=\frac{4}{15}$

The probabilities of a problem being solved by two students are $\frac{1}{2},\frac{1}{3}$. Then the probability of the problem being solved is

In a single throw of two dice, the probability of getting more than $7$ is

In each of the following experiments specify appropriate sample space A boy has a $1$ rupee coin, a $2$ rupee coin and a $5$ rupee coin in his pocket. He takes out two coins out of his pocket, one after the other.

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

Two dice are thrown. If first shows $5$, then the probability that the sum of the numbers appears on both is $8$ or more than $8$, is