Question

Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice ≤5.

State true or false: (give reason for your answer)
(i) A and B are mutually exclusive
(ii)A and B are mutually exclusive and exhaustive
(iii) A=B′
(iv)A and C are mutually exclusive
(v)A and B′ are mutually exclusive.
(vi) A′,B′,C are mutually exclusive and exhaustive.

Answer 1

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

(i) It is observed that A ∩ B = Φ
∴ A and B are mutually exclusive.
Thus, the given statement is true.

(ii) It is observed that A ∩ B = Φ and A ∪ B = S
∴ A and B are mutually exclusive and exhaustive.
Thus, the given statement is true.

(iii) It is observed that
B'={(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}=A
Thus, the given statement is true.

(iv) It is observed that A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠Φ
∴A and C are not mutually exclusive. Thus, the given statement is false.

(v) A∩B'=A∩A=A
∴A∩B'≠Φ
∴A and are not mutually exclusive. Thus, the given statement is false.

(vi) It is observed that A'∪B'∪C=S;
However, B'∩C={(2,1),(2,2),(2,3),(4,1)}≠Φ
Therefore, events A',B' and C are not mutually exclusive and exhaustive.
Thus, the given statement is false.