Types of Events

Types of Events

Events are subsets of the sample space, and the number of elements they contain determines their classification. The three fundamental types are impossible events, sure events, simple events, and compound events. Each plays a distinct role in describing what can or cannot happen in an experiment.

1. Impossible and Sure Events

The empty set $\phi$ and the full sample space $S$ are both valid events, but they represent two extremes.

Impossible event: An event that contains no sample point — that is, the event $\phi$. No outcome of the experiment can ever satisfy it.

Consider the experiment of rolling a single die. The sample space is $S = \{1, 2, 3, 4, 5, 6\}$. Let $E$ be the event "the number appearing on the die is a multiple of 7". Is there any number in $S$ that is a multiple of 7? No — 7, 14, 21, etc., are all outside the set $\{1, 2, 3, 4, 5, 6\}$. So the subset corresponding to $E$ is $\phi$. Since it is impossible for a die to show a multiple of 7, $E = \phi$ is called an impossible event.

Sure event: An event that contains every sample point — that is, the event $S$ itself. Every outcome of the experiment guarantees its occurrence.

Take the same die experiment. Let $F$ be the event "the number turns up is odd or even". Every integer from 1 to 6 is either odd or even, so $F = \{1, 2, 3, 4, 5, 6\} = S$. No matter what number appears, the event occurs. Hence $F = S$ is called a sure event (or certain event).

2. Simple Event

An event that contains exactly one sample point of the sample space is called a simple event (or elementary event).

If a sample space has $n$ distinct elements, there are exactly $n$ simple events — one for each sample point.

Example: Toss two coins. The sample space is $S = \{HH, HT, TH, TT\}$. The four simple events are:

• $E_1 = \{HH\}$
• $E_2 = \{HT\}$
• $E_3 = \{TH\}$
• $E_4 = \{TT\}$

Each of these contains only one outcome. No simple event can be broken down further into smaller events.

3. Compound Event

An event that contains more than one sample point is called a compound event.

Example: Toss a coin three times. The sample space $S$ has 8 elements:

Consider these events:

• $E$: 'Exactly one head appeared'
• $F$: 'At least one head appeared'
• $G$: 'At most one head appeared'

The subsets associated with these events are:

Each of these subsets contains more than one sample point. Therefore $E$, $F$, and $G$ are all compound events.

Key Distinctions

| Type | Number of sample points | Example (die roll) | …