Mutually Exclusive Events

The Idea of Mutually Exclusive Events

When you roll a fair die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$. Now consider two events:

• Event A: "an odd number appears" → $A = \{1, 3, 5\}$
• Event B: "an even number appears" → $B = \{2, 4, 6\}$

Can both A and B happen on the same roll? No — a single roll cannot be both odd and even. If A occurs, B cannot occur, and vice versa. In set language, $A \cap B = \varnothing$; the sets are disjoint.

This is the core idea: two events are mutually exclusive if the occurrence of one prevents the occurrence of the other. In other words, they cannot happen at the same time. The sets representing them have no common element.

A Contrast: Events That Are Not Mutually Exclusive

Stick with the same die experiment. This time consider:

• Event A: "an odd number appears" → $A = \{1, 3, 5\}$
• Event B: "a number less than 4 appears" → $B = \{1, 2, 3\}$

Now look at the intersection: $A \cap B = \{1, 3\}$. The outcome 3 belongs to both A and B. So if the die shows 3, both events happen simultaneously. Therefore, A and B are not mutually exclusive.

A Key Remark About Simple Events

A simple event is an event that contains exactly one outcome from the sample space. For example, in the die experiment, the event "the die shows a 4" is a simple event (the set $\{4\}$).

Simple events of a sample space are always mutually exclusive. Why? Because two different simple events correspond to two different outcomes, say $\{a\}$ and $\{b\}$ with $a \neq b$. Their intersection is $\{a\} \cap \{b\} = \varnothing$. No single outcome can belong to two different simple events at the same time.

Summary of the Key Ideas …