A picnic basket contains five jars: one of marmalade, two of peanut butter and two of jam. A boy removes one jar at random from the basket and then his sister takes two jars, both selected at random.

Draw up the probability distribution table for $J,$ the number of jars of jam selected by the sister, and show that $E\left(J\right)=0.8.$

Two ordinary fair dice are rolled. The product and the sum of the two numbers obtained are calculated. The score awarded, $S,$ is equal to the absolute (i.e. non-negative) difference between the product and the sum.
For example, if $5$ and $3$ are rolled, then $S=\left(5\times 3\right)-\left(5+3\right)=7.$

State the value of $S$ when $1$ and $4$ are rolled.

Two ordinary fair dice are rolled. The product and the sum of the two numbers obtained are calculated. The score awarded, $S,$ is equal to the absolute (i.e. non-negative) difference between the product and the sum.
For example, if $5$ and $3$ are rolled, then $S=\left(5\times 3\right)-\left(5+3\right)=7.$

Draw up a table showing the probability distribution for the $14$ possible values of $S,$ and use it to calculate $E\left(S\right)\mathit{.}$

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 $\mathrm{P}(2<\mathrm{X}<5)$.