Three vehicles from a company's six trucks, five vans, three cars and one motorbike are randomly selected and tested for roadworthiness.

Show that the probability of selecting three vans is $\frac{2}{91}.$

Three vehicles from a company's six trucks, five vans, three cars and one motorbike are randomly selected and tested for roadworthiness.

Draw up the probability distribution table for the number of vans 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 $X$.

In a game, a player attempts to hit a target by throwing three darts. With each throw, a player has a $30\%$ chance of hitting the target.

Draw up the probability distribution table for $H,$ the number of times the target is hit in a game.

Two students are randomly selected from a class of $12$ girls and $18$ boys.

Calculate the variance of the number of girls selected.

A sewing basket contains eight reels of cotton: four are green, three are red and one is yellow. Three reels of cotton are randomly selected from the basket.

Show that the expected number of yellow cotton reels is $0.375.$

A sewing basket contains eight reels of cotton: four are green, three are red and one is yellow. Three reels of cotton are randomly selected from the basket.
Find the expected number of red cotton reels.