1. Sample Space

In probability theory, we consider experiments as an abstraction for a thing that can happen: flipping a coin, rolling a dice, selecting a card from a deck of them, etc. The result, observation or measurement at the end of an experiment is called an outcome. We define a sample space $\Omega$ as the set of all possible outcomes that can result from an experiment. In other words, $\Omega$ contains all possiblities of an experiment that can occur.

Example: Consider the experiment of flipping a coin; it has a two possible outcomes. They are the coin landing on heads $H$ or on tails $L$. In this case, the sample space is simply $\Omega = ( H, T )$.

There are no limits on the cardinality of $\Omega$. However, we will use examples where it is finite. Generally a finite sample space $\Omega$ is defined as,

\[\Omega = ( \omega_1, \omega_2, ..., \omega_N )\]

where $\omega_j$ is the $j$-th elementary outcome: an indivisible outcome that cannot be broken down into smaller outcomes.

2. Events

An event $A$ is a subset of the sample space such that $A \subset \Omega$.