1: Sample spaces and events

Sample Spaces and Events

Probability theory provides a mathematical framework for analyzing uncertain phenomena. It begins with the concept of a random experiment—any process with uncertain outcomes that can be repeated under identical conditions (e.g., flipping a coin, rolling a die, measuring reaction time).

The sample space, denoted by $S$ or $\Omega$, is the fundamental building block. It is the set of all possible distinct outcomes of a random experiment. Each individual outcome is called a sample point or elementary outcome.

• Example 1 (Discrete): Flipping a coin: $S = \{\text{Heads}, \text{Tails}\}$.
• Example 2 (Discrete): Rolling a six-sided die: $S = \{1, 2, 3, 4, 5, 6\}$.
• Example 3 (Continuous): Measuring the lifetime (in hours) of a lightbulb: $S = \{x \mid x \geq 0\}$ (all non-negative real numbers).

Sample spaces can be discrete (containing a finite or countably infinite number of outcomes, like Examples 1 & 2) or continuous (containing an uncountably infinite number of outcomes, like Example 3).

An event is any subset of the sample space $S$. It represents a collection of outcomes of interest. An event is said to occur if the actual outcome of the experiment is any element within that subset.

• Simple Event: An event containing exactly one sample point (e.g., rolling a 3: $\{3\}$).
• Compound Event: An event containing two or more sample points (e.g., rolling an even number: $\{2, 4, 6\}$).

Since events are sets, set operations are crucial:

• Union ($A \cup B$): The event that occurs if $A$ occurs, $B$ occurs, or both occur (outcomes in $A$ or $B$ or both).
• Intersection ($A \cap B$): The event that occurs only if both $A$ and $B$ occur simultaneously (outcomes common to $A$ and $B$).
• Complement ($A^c$ or $A'$): The event that occurs if $A$ does not occur (all outcomes in $S$ not in $A$).
• Difference ($A \setminus B$): The event that occurs if $A$ occurs but $B$ does not (outcomes in $A$ but not in $B$).

Special types of events include:

• Certain Event: The sample space $S$ itself. It always occurs.
• Impossible Event: The empty set $\emptyset$. It contains no outcomes and never occurs.
• Mutually Exclusive (Disjoint) Events: Events $A$ and $B$ where $A \cap B = \emptyset$. They cannot occur simultaneously (e.g., rolling a 1 and rolling a 6 on a single die roll).

Understanding how to define sample spaces precisely and represent events using set notation is essential for applying probability axioms and rules effectively.