The probability distribution for the random variable $W$ is given in the following table.

$w$	$3$	$6$	$9$	$12$	$15$
$P\left(W=w\right)$	$2k$	$k^2$	$\frac{k}{2}$	$\frac{4}{5}-3k$	$\frac{13}{50}$

Explain why only one of your solutions is valid.

The probability distribution for the random variable $W$ is given in the following table.

Find $P\left(6⩽W<10\right)$

At a garden centre, there is a display of roses: $25$ are red, $20$ are white, $15$ are pink and $5$ are orange. Three roses are chosen at random. Show that the probability of selecting three red roses is approximately $0.0527.$

At a garden centre, there is a display of roses: $25$ are red, $20$ are white, $15$ are pink and $5$ are orange. Three roses are chosen at random.

Draw up the probability distribution table for the number of red roses selected.

At a garden centre, there is a display of roses: $25$ are red, $20$ are white, $15$ are pink and $5$ are orange. Three roses are chosen at random.

Find the probability that at least one red rose is selected.

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.

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

Draw the probability distribution table for van and find the probability that at most one van is selected.