The following table shows the probability distribution of a discrete random variable $X$.

$\mathbf{x}$	$\mathit{P}\mathbf{(}\mathbf{X}\mathbf{=}\mathbf{x}\mathbf{)}$
$-2$	$0.3$
$-1$	$\frac{1}{k}$
$0$	$\frac{2}{k}$
$1$	$0.1$
$2$	$0.1$

Find the value of $k$.

The following table shows the probability distribution of a discrete random variable $X$.

Find the expected value of $X$.

In a game a player rolls a biased tetrahedral (four-faced) die. The probability of each possible score is shown below.

Find the probability of a total score of six after two rolls.

A game involves spinning two spinners. One is numbered $1,2,3,4$. The other is numbered $2,2,4,4$. Each spinner is spun once and the number on each is recorded. Let $P$ be the product of the numbers on the spinners. Write down all the possible values for $P$.

A game involves spinning two spinners. One is numbered $1,2,3,4$. The other is numbered $2,2,4,4$. Each spinner is spun once and the number on each is recorded. Let $P$ be the product of the numbers on the spinners. Find the probability of each value of $P$.

A game involves spinning two spinners. One is numbered $1,2,3,4$. The other is numbered $2,2,4,4$. Each spinner is spun once and the number on each is recorded. Let $P$ be the product of the numbers on the spinners. Find the expected value of $P$.

A game involves spinning two spinners. One is numbered $1,2,3,4$. The other is numbered $2,2,4,4$. Each spinner is spun once and the number on each is recorded. Let $P$ be the product of the numbers on the spinners. A mathematician determines the amount of pocket money to give his son each week by getting him to play the game on Monday morning. If the son spins and the product is greater than $10$, then the boy gets $\pounds 10$. Otherwise, the boy gets $\pounds 5$. Find how much in total should the boy expect to get after $10$ weeks of playing the game.