Question

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:

1. A and B are mutually exclusive.
2. A and B are mutually exclusive and exhaustive events.
3. A and C are mutually exclusive events.
4. C and D are mutually exclusive and exhaustive events.
5. C, D and E are mutually exclusive and exhaustive events.
6. A' and B' are mutually exclusive events.
7. A, B, F are mutually exclusive and exhaustive events. 

Answer 1

When two dices are thrown, there are 62 = 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,

1. True, because A ∩ B = Φ
2. True, because A ∩ B = Φ and A ∪ B = S
3. False, because A ∩ C ≠  Φ
4. False, because C ∩ D = Φ but C ∪ D ≠ S
5. True, because C ∩ D ∩ E = Φ and C ∪ D ∪ E = S
6. True, because A' ∩ B' = Φ
7. False, because A ∩ B ∩ F ≠ Φ and A ∪ B ∪ F = S