Statistics and Probability - Sample Space Diagrams for Single Events

The Sample Space ($S$) is the set of all possible outcomes of a probability experiment. For a single event, it is often visually represented as a list within curly brackets, such as $S = \{1, 2, 3, 4, 5, 6\}$ for a standard die, or as a set of points on a number line.

The Probability Scale is a visual representation of the likelihood of an event, ranging from $0$ (Impossible) to $1$ (Certain). A probability of $0.5$ (or $\frac{1}{2}$) represents an 'Even Chance,' such as flipping a tail on a fair coin.

Equally Likely Outcomes occur when every possible result in the sample space has the same chance of occurring. Visually, this is represented by a fair spinner divided into equal-sized sectors or a balanced die where each face has the same surface area.

An Event is a specific outcome or a collection of outcomes from the sample space. For example, in the sample space of a die, 'rolling an even number' is an event consisting of the subset $\{2, 4, 6\}$.

The Complement of an Event ($A'$) includes all outcomes in the sample space that are not part of event $A$. If you visualize the sample space as a box (Venn Diagram), if event $A$ is a circle inside, the complement $A'$ is the entire area outside that circle but within the box.

Theoretical Probability is determined by mathematical reasoning rather than physical experiments. It assumes that physical objects like coins and dice are 'fair' and 'unbiased,' meaning their visual and physical symmetry ensures equal probability for all faces.

The Sum of Probabilities for all possible mutually exclusive outcomes in a single-event sample space must always equal $1$. This can be visualized as a pie chart where the sum of all individual slices accounts for the whole circle.