A pack of five $DVDs$ contains three movies and two documentaries. Three $DVDs$ are selected and the following table shows the probability distribution for $M,$ the number of movies selected.

$m$	$1$	$2$	$3$
$P\left(M=m\right)$	$0.3$	$0.6$	$0.1$

Draw up the probability distribution table for $D,$ the number of documentaries selected.

A fair $4$ -sided die, numbered $1, 2, 3$ and $5,$ is rolled twice. The random variable $X$ is the sum of the two numbers on which the die comes to rest.

Show that $P\left(X=8\right)=\frac{1}{8}$

A fair $4$ -sided die, numbered $1, 2, 3$ and $5,$ is rolled twice. The random variable $X$ is the sum of the two numbers on which the die comes to rest.

Draw up the probability distribution table for $X$.

There are eight letters in a post box, and five of them are addressed to Mr Nut. Mr Nut removes four letters at random from the box.

Find the probability that none of the selected letters are addressed to Mr Nut.

There are eight letters in a post box, and five of them are addressed to Mr Nut. Mr Nut removes four letters at random from the box.

Draw up the probability distribution table for $N,$ the number of selected letters that are addressed to Mr Nut.

There are eight letters in a post box, and five of them are addressed to Mr Nut. Mr Nut removes four letters at random from the box.

Describe one significant feature of a vertical line graph or bar chart that could be used to represent the probability distribution for $N.$

$\theta$ is a discrete random variable and $\theta \in \left(3, 4, 5, 6\right).$ Given that $P\left(\theta =q\right)=c{q}^2$ find the value of the constant $c.$