Sample Space and Events

To apply probability, we first need to describe the structure of possible outcomes.

This begins with the sample space, denoted \( \Omega \), which is the set of all possible outcomes of an experiment. For example, when rolling a standard six-sided die, the sample space is

\[ \Omega = \{1, 2, 3, 4, 5, 6\} \]

An event is any subset of this sample space. Events can be simple (a single outcome, like rolling a 3, i.e., \( \{3\} \)), or compound (multiple outcomes, like rolling an even number, \( \{2, 4, 6\} \)).

Events can also be mutually exclusive, meaning they cannot occur at the same time (like \( \{1\} \) and \( \{2\} \) on a single roll), or exhaustive, meaning they together account for all possible outcomes (e.g., \( \{1, 2\} \) and \( \{3, 4, 5, 6\} \) are exhaustive over \( \Omega \)).

Defining events clearly allows us to apply probability rules effectively and consistently in both simple and complex situations.