Q.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 $\leq 5$. Describe the events

Concept: Set Operations on events. Two dice give $36$ ordered pairs. Here $A$ = even first die ($18$ outcomes), $B$ = odd first die ($18$), $C$ = sum $\le 5$ ($10$).

• (i) $A'$: odd first die $=B$.
• (ii) not $B$: even first die $=A$.
• (iii) $A\cup B$: even or odd first die $=S$ (all $36$).
• (iv) $A\cap B$: $\varnothing$ (first die can't be both).
• (v) $A\cap C'$ (A but not C): even first die and sum $>5$ — $18-4=14$ outcomes.
• (vi) $B\cup C$: $|B|+|C|-|B\cap C|=18+10-6=22$ outcomes.
• (vii) $B\cap C$: odd first die and sum $\le 5$ $=\{(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)\}$ — $6$ outcomes. …

When two dice are thrown, each outcome is an ordered pair $(i,j)$ with $i$ from the first die and $j$ from the second. There are $6\times 6=36$ equally likely outcomes.

The three given events

$A$ — even number on the first die ($i\in\{2,4,6\}$): $|A|=18$.

$B$ — odd number on the first die ($i\in\{1,3,5\}$): $|B|=18$.

$C$ — sum $\le 5$:

Describing each requested event

(i) $A'$ — "not even on the first die" means odd on the first die, so $A'=B$ (an odd number on the first die).

(ii) not $B$ — "not odd on the first die" means even on the first die, so $B'=A$.

(iii) $A$ or $B$ $=A\cup B$ — the first die is even or odd, which is always the case, so $A\cup B=S$ (all $36$ outcomes).

(iv) $A$ and $B$ $=A\cap B$ — the first die cannot be even and odd at once, so $A\cap B=\varnothing$.

(v) $A$ but not $C$ $=A\cap C'$ — even first die and sum $>5$. Remove from $A$ the outcomes with sum $\le 5$, namely $(2,1),(2,2),(2,3),(4,1)$:

which has $18-4=14$ outcomes. …