Statistics and Probability - Calculating Probability of Events

The Probability Scale: Probability is the numerical measure of the likelihood that an event will occur, ranging from $0$ to $1$. Visualize this as a horizontal line where $0$ (on the far left) represents an impossible event, $0.5$ (in the middle) represents an even chance, and $1$ (on the far right) represents a certain event. All probabilities are expressed as fractions, decimals, or percentages within this range.

Sample Space ($S$): The sample space is the set of all possible outcomes of an experiment. It can be visualized as a list within curly brackets, such as $S = \{1, 2, 3, 4, 5, 6\}$ for a fair die, or as a grid where the x-axis and y-axis represent two different trials (like rolling two dice) to show every possible coordinate combination.

Theoretical vs. Experimental Probability: Theoretical probability is based on reasoning and mathematical calculation of what 'should' happen. Experimental probability (or relative frequency) is based on actual data from trials. You can visualize experimental probability using a frequency table that records the number of times an outcome occurs divided by the total number of trials conducted.

Complementary Events: The complement of event $A$, denoted as $A'$, consists of all outcomes in the sample space that are NOT in $A$. In a Venn diagram, if event $A$ is a circle inside a rectangular box (the sample space), $A'$ is the entire shaded area outside that circle but inside the box. The sum of the probability of an event and its complement is always $1$.

Tree Diagrams: A visual tool used to calculate probabilities for multiple events occurring in sequence. Each stage of the experiment is represented by a set of 'branches'. You multiply probabilities along the branches to find the probability of a specific path (e.g., flipping a Head then a Tail) and ensure the sum of probabilities at any set of branches equals $1$.

Venn Diagrams and Two-Way Tables: These are used to represent the relationship between two or more events. In a Venn diagram, the overlapping section (the intersection) represents outcomes where both events occur ($A \cap B$), while the area covered by both circles (the union) represents outcomes where either or both events occur ($A \cup B$). Two-way tables organize this same data into rows and columns for easy summation.

Mutually Exclusive Events: Events that cannot happen at the same time. Visually, in a Venn diagram, mutually exclusive events are represented by two circles that do not overlap or touch. If events are mutually exclusive, the probability of either occurring is simply the sum of their individual probabilities.