Question

Two dice are thrown. The events A, B, C, D, E and F are describes as follows. 

A = Getting an even number on the first die. 

B. Getting an odd number on the first die. 

C = Getting at most 5 as sum of the numbers on the two dice. 

D = Getting the sum of the numbers on the dice greater than 5 but less than 10. 

E = Getting at least 10 as the sum of the numbers on the dice. 

F = Getting an odd number on one of the dice. 

(i) Describe the following events: A and B, B or C, B and C, A and E, A or F, A and F 

(ii) State true or false:

(a) A and B are mutually exclusive.

(b) A and B are mutually exclusive and exhaustive events 

(c) A and C are mutually exclusive events 

(d) C and D are mutually exclusive and exhaustive events 

(e) C, d and E are mutually exclusive and exhaustive events 

(f) A’ and B’ are mutually exclusive events 

(g) A, B, F are mutually exclusive and exhaustive events

Answer 1

To Find: Describe the given events. 

Explanation: when two dice are thrown then the no. of possible outcomes are 6² = 36 

Now, According to the question, 

A = Getting an even number of the first die. 

A = {(2, 1)(2, 2)(2, 3)(2, 4)(2, 5)(2, 6)(4, 1)(4, 2)(4, 3)(4, 4)(6, 1)(6, 2)(6, 3)(6, 4)(6, 5)(6, 6)} 

B = Getting an odd number on the first dice 

B = {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5)(5, 6)} 

C = Getting at most 5 as the sum of numbers on the two dices. 

C = {(1, 1)(1, 2)(1, 3)(1, 4)(2, 1)(2, 2)(2, 3)(3, 1)(3, 2)(4, 1)} 

D = Getting a sum greater than 5 but less than 10 

D = {(1, 5)(1, 6) (2, 4)(2, 5) (2, 6)(3, 3)(3, 4)(3, 5)(3, 6)(4, 2)(4, 3)(4, 4)(4, 5)(5, 1) (5, 2)(5, 3) (5, 4)(6, 1) (6, 2) (6, 3)} 

E = Getting at least 10 as the sum of numbers on the dices 

{(4, 6)(5, 5)(5, 6)(6, 4)(6, 5)(6, 6)} 

F = Getting an odd number on one of the dices 

{(1, 2)(1, 4)(1, 6)(2, 1)(2, 3)(2, 5)(3, 2)(3, 4)(3, 6)(4, 1)(4, 3)(4, 5)(5, 2)(5, 4)(5, 6)(6, 1)(6, 3)(6, 5) Now, 

(i) A and B = AՌB = Փ 

Since, There is no common events in Both A and B so the intersection is Null (Փ). 

⇒ B or C = BUC 

BUC = {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)(2, 1)(2, 2)(2, 3)(4, 1)} 

⇒ B and C = BՌC 

BՌC = {(1, 1), (1, 2)(1, 3)(1, 4)(3, 1)(3, 2)} 

⇒ A and E = AՌE 

AՌE = {(4, 6)(6, 4)(6, 5)(5, 6)(6, 6)} 

⇒ A or F = AUF 

AUF = {(2, 1)(2, 2)(2, 3)(2, 4)(2, 5)(2, 6)(4, 1)(4, 2)(4, 3)(4, 4)(6, 1)(6, 2)(6, 3)(6, 4)(6, 5)(6, 6)(1, 2)(1, 4)(1, 6) (3, 2)(3, 4)(3, 6)(4, 5)(5, 2)(5, 4)(5, 6)}

(ii) 

(a) True, because AՌB = Փ 

(b) True, because AՌB = Փ and AUB = S 

(c) False, because AՌC = Փ 

(d) False, because CՌD = Փ but CUD≠S 

(e) True, because CՌDՌE = Փ and CUDUE = S 

(f) True, because A’ՌB’ = Փ 

(g) False, because AՌBՌF = Փ and AUBUF = S