Q.On a single roll of a fair die, let $E_1 =$ 'getting an even number' and $E_2 =$ 'getting an odd number'. Examine whether $E_1$ and $E_2$ are

Check the intersection for mutual exclusivity and the union for exhaustiveness, separately.

For a single die roll, $S=\{1,2,3,4,5,6\}$. Here $E_1 = \{2,4,6\}$ and $E_2=\{1,3,5\}$.

Mutual exclusivity check: $E_1 \cap E_2$ — is any number both even and odd? No number satisfies both, so $E_1 \cap E_2 = \emptyset$. By the definition in §3, $E_1$ and $E_2$ are mutually exclusive.

Collective exhaustiveness check: $E_1 \cup E_2 = \{2,4,6\} \cup \{1,3,5\} = \{1,2,3,4,5,6\}$, which is exactly $S$. By the definition in §3, $E_1$ and $E_2$ are collectively exhaustive. …

Students sometimes assume mutual exclusivity automatically implies collective exhaustiveness (or vice versa) because this example happens to show both — but the two properties are independent and must always be checked separately (e.g. 'drawing a heart' and 'draw …