Q.Events E and F are such that P(not E or not F) = 0.25. State whether E and F are mutually exclusive.

Mutually Exclusive Events

The "Can't Happen Together" Idea

Roll a single six-sided die and look at two events:

• Event $A$: the die shows an even number — $A = \{2, 4, 6\}$.
• Event $B$: the die shows an odd number — $B = \{1, 3, 5\}$.

Can one roll be both even and odd at the same time? No — not a single outcome belongs to both. Events like these, which can never occur together in the same trial, are called mutually exclusive.

Now compare a different pair:

• Event $A$: an even number — $\{2, 4, 6\}$.
• Event $C$: a number greater than 3 — $\{4, 5, 6\}$.

Here $4$ and $6$ belong to both, so $A$ and $C$ can happen together. They are not mutually exclusive.

The Precise Definition

The Addition Rule for Mutually Exclusive Events

This is the property most often used in the exam. Because there is no overlap to double-count, the general addition rule

collapses. Since $A \cap B = \varnothing$, we have $P(A \cap B) = 0$, so:

For three mutually exclusive events $A$, $B$, $C$ (every pair disjoint):

More Than Two Events

A collection of events $E_1, E_2, \ldots, E_n$ is called mutually exclusive if every pair among them is mutually exclusive, i.e. $E_i \cap E_j = \varnothing$ for all $i \neq j$. For instance, when a die is rolled the six simple events $\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}$ are all mutually exclusive — no two of them can happen on the same roll.

Worked Examples

Example 1 — Drawing one card from a standard deck.

• Event $A$: the card is a heart.
• Event $B$: the card is a spade.

A single card cannot be both a heart and a spade, so $A \cap B = \varnothing$: these are mutually exclusive.

Example 2 — Drawing one card from a standard deck.

• Event $A$: the card is a heart.
• Event $B$: the card is a king.

The king of hearts lies in both events, so $A \cap B = \{\text{king of hearts}\} \neq \varnothing$: these are not mutually exclusive. …