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

Given:

Two dice are thrown.

A: Getting an even number on the first die = {2, 4, 6}
B: Getting an odd number on the first die = {1, 3, 5}
C: Getting sum of numbers on the dice ≤ 5

---

(i) A and B are mutually exclusive

An outcome cannot be both even and odd on the first die at the same time.

True

---

(ii) A and B are mutually exclusive and exhaustive

Every outcome on the first die is either even or odd.

So A ∪ B = Sample Space of first die.

True

---

(iii) A = B′

B′ means “not odd on the first die”, i.e. even.

So B′ = A

True

---

(iv) A and C are mutually exclusive

Example: (2,1) → first die is even and sum = 3 ≤ 5.

So A and C can occur together.

False

---

(v) A and B′ are mutually exclusive

From (iii), B′ = A

So A and B′ are the same event, not mutually exclusive.

False

---

(vi) A′, B′, C are mutually exclusive and exhaustive

A′ = odd on first die = B

B′ = even on first die = A

C overlaps with both A and B (e.g. (1,1), (2,1)).

Hence they are neither mutually exclusive nor exhaustive.

False

---

Final Answers:

(i) True
(ii) True
(iii) True
(iv) False
(v) False
(vi) False