A box contains 15 balls numbered 1, 2,…, 15. The balls numbered 1–5 are white and those numbered 6–15 are black. We select a ball at random, and record its color and the number on it.

(i) Write down a suitable sample space for this experiment.

(ii) We define the events Ai: the ball selected has a number less than or equal to i, for 1 ≤ i ≤ 15;

Bi: the ball selected has a number greater than or equal to i on it, for 1 ≤ i ≤ 15;

C: the selected ball is white;

D: the selected ball is black.

Examine which of the following assertions are correct and which are false:

(a) A5 = C;

(b) A4 ⊆ C;

(c) AiBi = ∅ for any i = 1, 2,…, 15;

(d) Ai−1Bi = ∅ for any i = 2, 3,…, 15;

(e) AiBi+1 = ∅ for any i = 1, 2,…, 14;

(f) the events C and D are mutually exclusive;

(g) A10B5 ⊆ C;

(h) A7D = ∅;
(i) A′
5 = D;
(j) Ai ∪ Bi+1 = Ai for any i = 1, 2,…, 14;
(k) A1 ⊆ A2 ⊆ · · · ⊆ A15;
(l) Ai ∪ Bi = Ω for any i = 1, 2,…, 15;
(m) B1 ⊆ B2 ⊆ · · · ⊆ B15;
(n) Ai ∪ Bi+1 = ∅ for any i = 1, 2,…, 15;
(o) A′i = Bi+1 for i = 1, 2,…, 14;
(p) (A10 − C)B6 = ∅;
(q) (A12 − D) ⊆ B5;
(r) D − B11 = A10 − A5.