The probability that a boy succeeds with each basketball shot is $\frac{7}{9}.$ He takes two shots and the discrete random variable $S$ represents the number of successful shots. Show that $P\left(S=0\right)=\frac{4}{81}$ and draw up the probability distribution table for $S.$

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.

A pack of five $DVDs$ contains three movies and two documentaries. Three $DVDs$ are selected and the following table shows the probability distribution for $M,$ the number of movies selected.

Draw up the probability distribution table for $D,$ the number of documentaries selected.