A bag contains 12 red marbles, 8 blue marbles and 4 green marbles. The blue marbles are numbered 1 to 8, the red marbles are numbered from 9 to 20 and green marbles are numbered from 21 to 24. (a) A handful of 6 marbles are taken from the bag and placed on a table. Let X be the number of blue marbles on the table. i. Name the most appropriate distribution to use to model X and the parameter(s) of the distribution. [2 marks] ii. Find E(X ) and sd(X) [3 marks] iii. What is the probability that 5 of the handful of 6 marbles are not blue? [2 marks] (b) Suppose that the marbles are drawn out one at a tine and their number noted. They are then put back in the bag and the bag is shaken. Let Y be the number of draws before the first multiple of 4 is drawn. i. Name the most appropriate distribution to use to model Y and the parameter(s) of the distribution. [2 marks] ii. What is the probability that the first multiple of 4 is drawn at the third draw? [2 marks] iii. Given that the first draw had done and the number shown was not a multiple of 4, what is the probability that the first multiple of 4 is drawn on the third draw. [2 marks] iv. What is the expected number of draws before a first multiple of 4 is drawn? [1 marks] Let A and B be events on the same sample space. Define the event C to be the set of all elements of B that are not in A. Show that if A C B then, P(C) = P(B) - P(A)