Problem

A bag contains a large number of beads of the same size and shape. The beads are either black or white.

Black beads and white beads occur in the ratio 2 : 5 respectively.

In a game a player takes a random sample of 3 beads from the bag.

The player scores

• 4 points for each black bead taken
• 1 point for each white bead taken

The random variable $X$ represents the total number of points for the 3 beads.

(a) Find the sampling distribution of $X$

A random sample of $n$ sets of 3 beads is taken. The random variable $Y$ represents the number of these $n$ sets that have a total of exactly 9 points.

(b) Calculate the minimum value of $n$ such that $P(Y \geqslant 1) > 0.95$

(Total for Question 4 is 9 marks)