The faces of a biased die are numbered $1, 2, 3, 4, 5$ and $6.$ The random variable $X$ is the score when the die is thrown. The probability distribution table for $X$ is given.

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$P\left(X=x\right)$	$p$	$p$	$p$	$p$	$0.2$	$0.2$

The die is thrown $3$ times. Find the probability that the score is at least $4$ on at least $1$ of the $3$ throws.

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.

Find the probability that the sister selects her jars from a basket that contains:

exactly one jar of jam

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.

Find the probability that the sister selects her jars from a basket that contains:

exactly two jars of jam.

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.