Occurrence of an Event

Occurrence of an Event

When we talk about an event in probability, we are not just naming a possibility — we are describing a specific collection of outcomes from the sample space. The real question is: when does that event actually happen?

Consider the experiment of throwing a fair die. The sample space is $S = \{1, 2, 3, 4, 5, 6\}$. Let $E$ denote the event "a number less than 4 appears". This means $E = \{1, 2, 3\}$.

Now suppose you throw the die and the outcome is 1. Since 1 belongs to $E$, we say that event $E$ has occurred. Similarly, if the outcome is 2 or 3, event $E$ has occurred. But if the outcome is 4, 5, or 6, then the outcome does not belong to $E$, and we say that event $E$ has not occurred.

This leads to the precise definition:

The idea is simple but foundational: an event occurs exactly when the actual outcome is one of the outcomes that the event includes. Nothing more, nothing less.

This definition applies to every event, whether it is a simple event (a single outcome) or a compound event (a collection of outcomes). For example, if $F$ is the event "an even number appears", then $F = \{2, 4, 6\}$. If the die shows 4, $F$ has occurred; if it shows 3, $F$ has not occurred.

The same logic extends to the complement of an event. If $E$ has not occurred, then the outcome $\omega$ belongs to the complement $E'$ (or $E^c$), which is $S - E$. So saying "event $E$ has not occurred" is exactly the same as saying "the complementary event $E'$ has occurred". …