Mutually Exclusive and Collectively Exhaustive Events

Two or more events are called mutually exclusive if they cannot occur together in a single performance of the experiment — that is, no sample point is common to any two of them. In set notation, events $E_1$ and $E_2$ are mutually exclusive if $E_1 \cap E_2 = \emptyset$ (their intersection is empty).

A collection of events is called collectively exhaustive if, taken all together, they cover EVERY possible outcome of the sample space — their union equals $S$ itself: $E_1 \cup E_2 \cup \cdots = S$.

Worked illustration on a single die roll: let $E_1 =$ "getting an even number" $= \{2,4,6\}$ and $E_2 =$ "getting an odd number" $=\{1,3,5\}$.

• $E_1 \cap E_2 = \emptyset$ (no number is both even and odd) — so $E_1$ and $E_2$ are mutually exclusive.
• $E_1 \cup E_2 = \{1,2,3,4,5,6\} = S$ (every possible die outcome is either even or odd) — so $E_1$ and $E_2$ are also collectively exhaustive.

These two properties are independent of each other — an event pair can have one without the other:

• "Drawing a heart" (13 cards) and "drawing a spade" (13 cards) from a deck ARE mutually exclusive (no card is both a heart and a spade), but they are NOT collectively exhaustive — together they account for only 26 of the 52 cards, leaving every diamond and every club uncovered.
• "Getting a number less than 4" $=\{1,2,3\}$ and "getting an even number" $=\{2,4,6\}$ on a die are NOT mutually exclusive (they share the sample point $\{2\}$), even though together they cover $\{1,2,3,4,6\}$ — and since $5$ is missing, they are not collectively exhaustive either. …