Exhaustive Events

Exhaustive Events

When we throw a die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$. Consider three events:

• $A$: 'a number less than 4 appears' → $A = \{1, 2, 3\}$
• $B$: 'a number greater than 2 but less than 5 appears' → $B = \{3, 4\}$
• $C$: 'a number greater than 4 appears' → $C = \{5, 6\}$

Now take the union of these three events:

The union of $A$, $B$ and $C$ covers the entire sample space. Events with this property are called exhaustive events.

In simpler terms, events are exhaustive if at least one of them must necessarily occur whenever the experiment is performed. No outcome of the experiment falls outside all the events — every possible result is covered by at least one event.

Mutually Exclusive and Exhaustive Events

When events are both pairwise disjoint (mutually exclusive) and their union equals the sample space, they form a special combination.

This means the events partition the sample space into non-overlapping pieces that together cover everything. Every outcome belongs to exactly one of these events. …