A fair triangular spinner has sides labelled $0,1$ and $2,$ and another fair triangular spinner has sides labelled $-1,0$ and $1.$ The score, $X,$ is equal to the sum of the squares of the two numbers on which the spinners come to rest.

List the five possible values of $X.$

A fair triangular spinner has sides labelled $0,1$ and $2,$ and another fair triangular spinner has sides labelled $-1,0$ and $1.$ The score, $X,$ is equal to the sum of the squares of the two numbers on which the spinners come to rest.

Draw up the probability distribution table for $X.$

A fair triangular spinner has sides labelled $0,1$ and $2,$ and another fair triangular spinner has sides labelled $-1,0$ and $1.$ The score, $X,$ is equal to the sum of the squares of the two numbers on which the spinners come to rest.

Given that $X<4,$ find the probability that a score of $1$ is obtained with at least one of the spinners.

A fair triangular spinner has sides labelled $0,1$ and $2,$ and another fair triangular spinner has sides labelled $-1,0$ and $1.$ The score, $X,$ is equal to the sum of the squares of the two numbers on which the spinners come to rest.

Find the exact value of $a,$ such that the standard deviation of $X$ is $\frac{1}{a}\times E\left(X\right).$

A discrete random variable $X,$  where $X\in \left\{2, 3, 4, 5\right\}$, is such that $P\left(X=x\right)=\frac{{\left(b-x\right)}^2}{30}.$ Calculate the two possible values of $b.$

A discrete random variable $X,$  where $X\in \left\{2, 3, 4, 5\right\}$, is such that $P\left(X=x\right)=\frac{{\left(b-x\right)}^2}{30}.$ Hence, find $\mathrm{P}(2<\mathrm{X}<5)$.

Set $A$ consists of the ten digits $0, 0, 0, 0, 0, 0, 2, 2, 2, 4$
Set $B$ consists of the seven digits $0, 0, 0, 0, 2, 2, 2$
One digit is chosen at random from each set. The random variable $X$ is defined as the sum of these two digits.

Show that $P(X=2)=\frac{3}{7}.$

Set $A$ consists of the ten digits $0, 0, 0, 0, 0, 0, 2, 2, 2, 4$
Set $B$ consists of the seven digits $0, 0, 0, 0, 2, 2, 2$
One digit is chosen at random from each set. The random variable $X$ is defined as the sum of these two digits.

Tabulate the probability distribution of $X.$

Set $A$ consists of the ten digits $0, 0, 0, 0, 0, 0, 2, 2, 2, 4$
Set $B$ consists of the seven digits $0, 0, 0, 0, 2, 2, 2$
One digit is chosen at random from each set. The random variable $X$ is defined as the sum of these two digits.

Find $E\left(X\right)$ and Var $\left(X\right).$