Q.Refer to question 6 above, state true or false: (give reason for your answer)

Concept: Mutually Exclusive and Exhaustive Events. From question 6 (two dice): $A$ = even first die, $B$ = odd first die, $C$ = sum $\le 5$. Mutually exclusive means $A\cap B=\varnothing$; exhaustive means the union is the whole sample space $S$.

• (i) $A\cap B=\varnothing$ (first die can't be both even and odd). True.
• (ii) They are mutually exclusive and $A\cup B=S$ (every first die is even or odd), so also exhaustive. True.
• (iii) $B'$ = not-odd first die = even first die $=A$. True.
• (iv) $A\cap C=\{(2,1),(2,2),(2,3),(4,1)\}\ne\varnothing$. False. …

This question refers to the events of question 6, where two dice are thrown ($36$ equally likely outcomes) and

• $A$ = an even number on the first die ($i\in\{2,4,6\}$, $18$ outcomes),
• $B$ = an odd number on the first die ($i\in\{1,3,5\}$, $18$ outcomes),
• $C$ = the sum of the numbers $\le 5$, i.e. $C=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1)\}$.

Two definitions we will use: two events are mutually exclusive if they cannot occur together ($A\cap B=\varnothing$), and a collection of events is exhaustive if together they cover the whole sample space (their union is $S$).

(i) $A$ and $B$ are mutually exclusive. The first die cannot be even and odd at the same time, so $A\cap B=\varnothing$. True.

(ii) $A$ and $B$ are mutually exclusive and exhaustive. They are mutually exclusive (part i), and every outcome has a first die that is either even or odd, so $A\cup B=S$. Being disjoint and covering $S$, they are also exhaustive. True.

(iii) $A=B'$. $B'$ means "not an odd first die", i.e. an even first die, which is exactly $A$. So $A=B'$. True. …