An experiment consists of recording boy-girl composition of families with $2$ children. What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births ?

A coin is tossed three times, consider the following events.

$A: $ ' No head appears ',  $B:$ ' Exactly one head appears ' and  $C:$ ' Atleast two heads appear '

Do they form a set of mutually exclusive and exhaustive events?

A card is selected from a pack of $52$ cards. Calculate the probability that the card is an ace of spades.

The probability of happening an event $A$ in one trial is $0.4$. The probability that the event $A$ happens at least once in three independent trials is

A pair of a dice thrown, if $5$ appears on at least one of the dice, then the probability that the sum is $10$ or greater is

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

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