Basic Concepts

An experiment in probability is a process or action with uncertain results, whose outcomes can be observed or measured.

An outcome is a single possible result of an experiment, such as rolling a $4$ on a die or getting heads on a coin flip.

The sample space, denoted \( \Omega \) (

$\Omega$

: the Greek capital letter "Omega"), is the set of all possible outcomes of an experiment; for example, for a die roll, \( \Omega = \{1, 2, 3, 4, 5, 6\} \).

For a coin flip, \( \Omega = \{\text{H}, \text{T}\} \), and for flipping two coins, \( \Omega = \{\text{HH}, \text{HT}, \text{TH}, \text{TT}\} \).

Experiments must be clearly defined to properly construct sample spaces and ensure accurate probability modeling.

Sample spaces may be finite, countably infinite (similar to the infinity of $\mathbb{N}$: the set of integers,) or uncountable (similar to the vast infinity of $\mathbb{R}$: the set of real numbers,) but in discrete probability, we focus on the first two types.