Problem

A bag contains a large number of counters with one of the numbers 5, 10 or 20 written on each of them in the ratio 5 : 2 : a A jar contains a large number of counters with one of the numbers 5 or 10 written on each of them in the ratio 1 : 3 One counter is selected at random from the bag and then two counters are selected at random from the jar. The random variable $R$ represents the range of the numbers on the 3 counters.

Given that $P(R = 15) = \dfrac{63}{256}$

(a) by forming and solving an equation in $a$, show that $a = 9$

(b) find the sampling distribution of $R$

(Total for Question 6 is 9 marks)