Two dice are thrown. The events *A*, *B*, *C*, *D*, *E* and *F* are described as :*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.

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.

#### Solution

When two dices are thrown, there are 6^{2} = 36 possible outcomes.*A *= Getting an even number on the first dice

= {(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* = Getting an odd number on the first dice

= {(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 the numbers on the two dices.

= {(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

= {(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 the 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,

(a) True, because A ∩ B = Φ

(b) True, because A ∩ B = Φ and A ∪ B = S

(c) False, because A ∩ C ≠ Φ

(d) False, because C ∩ D = Φ but C ∪ D ≠ S

(e) True, because C ∩ D ∩ E = Φ and C ∪ D ∪ E = S

(f) True, because A' ∩ B' = Φ

(g) False, because A ∩ B ∩ F ≠ Φ and A ∪ B ∪ F = S