Consider the set of all $7-$digit numbers formed by the digits $0,1,2,3,4,5,6$, each chosen exactly once. If a number is randomly drawn from this set, the probability that it is divisible by $4$ is

One card is drawn from a well shuffled deck of $52$ cards. If each outcome is equally likely, calculate the probability that the card will be not a black card.

In a relay race there are five teams $A, \,B, \,C, \,D$ and $E$. What is the probability that $A, \,B$ and $C$ finish first, second and third, respectively.

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

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

A box containing $4$ white pens and $2$ black pens. Another box containing $3$ white pens and $5$ black pens. If one pen is selected from each box, then the probability that both the pens are white is equal to

Find the probability that the two digit number formed by digits $1, 2, 3, 4, 5$ is divisible by $4$ (while repetition of digit is allowed)