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

Describe the events

1. A' 
2. not B
3. A or B
4. A and B
5. A but not C
6. B or C
7. B and C
8. A ∩ B' ∩ C'

Answer 1

Sample space when two coins are tossed

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

A = Getting an even number on the first die.

= {(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

B = Getting an odd number on the first die.

= {(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 the sum of the numbers on the dice ≤ 5

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

i. A’ = S – A

= {(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)}

= B

ii. Not B = B’ = No odd number on the first die

= {(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

iii. A or B = A ∪ B = {x : x is an even number on the first die} ∪ {is an odd number on the first die}
= S

iv. A and B = A ∩ B

= {x : x is an even number on the first die} ∩ {x is an odd number on the first die}

= ϕ

v. A but not C

= {x : x is an even number on the first dice} – {sum of numbers obtained on the dice ≤ 5}

A – C = {(2, 1), (2, 2), …(2, 6), (4, 1), (4, 2), …(4, 2), …(4, 6), (6, 1), (6, 2), ….(6, 6)} – {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
= {(2, 4), (2, 5), (2, 6), (4, 2), (4, 3), …(4, 6), (6, 1), (6, 2), .…(6, 6)}

vi. B or C = B ∪ C = {x : x will be an odd number on the first die} ∪ {Sum of numbers obtained on dice ≤ 5} 

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

= {(1, 1), (1, 2), …(1, 6), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), …(3, 6), (4, 1), (5, 1), (5, 2), (5, 3), …(5, 6).

viii. B and C i.e. B ∩ C = {(1, 1), …(1, 6), (3, 1), (3, 2), …(3, 6), (5, 1), (5, 2), (5, 3), …(5, 6) ∩ {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2) (2, 3), (3, 1), (3, 2), (4, 1)}

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

viii. Here, B’ = A

∴ A ∩ B’ = A ∩ A = A

∴ A ∩ B’ ∩ C’ = {(2, 1), (2, 2), …(2, 6), (4, 1), (4, 2), …(4, 6), (6, 1), (6, 2), …(6, 6)} ∩ {(1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 2), (4, 3), …(4, 6), (5, 1), (5, 2), …(5, 6), (6, 1), (6, 2), …(6, 5)}

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