Statistics and Probability - Introduction to Probability

The Probability Scale: Probability is a measure of the likelihood that an event will occur, represented on a scale from $0$ to $1$. Visually, this is a horizontal line where $0$ (impossible) is on the far left, $0.5$ (even chance) is in the middle, and $1$ (certain) is on the far right. Values closer to $1$ represent 'likely' events, while values closer to $0$ represent 'unlikely' events.

Theoretical Probability: This is calculated based on reasoning and the assumption that all outcomes are equally likely. For example, when looking at a standard six-sided die, we assume each face has a $1$ in $6$ chance. The sample space is the set of all possible outcomes, often listed in curly brackets as $\{1, 2, 3, 4, 5, 6\}$.

Experimental Probability (Relative Frequency): This is based on actual data or experiments. It is calculated after performing trials. For instance, if you flip a coin $100$ times and get heads $55$ times, the experimental probability is $0.55$. As the number of trials increases, the experimental probability usually gets closer to the theoretical probability.

Sample Space Diagrams: For experiments involving two steps (like rolling two dice), a sample space diagram or a 2D grid is used. Imagine a square grid where the horizontal axis represents the outcomes of the first die ($1-6$) and the vertical axis represents the second die ($1-6$). Each intersection point in the grid represents a unique combined outcome, such as $(2, 3)$, totaling $36$ possible outcomes.

Complementary Events: The complement of an event $A$ is the event that $A$ does not happen, denoted as $A'$. Visually, if you imagine a Venn diagram where a circle represents event $A$ inside a rectangular box (the universal set), the area outside the circle but inside the box represents $A'$. The sum of the probability of an event and its complement is always $1$.

Tree Diagrams: These are used to visualize probabilities of multi-stage events. Each stage of the experiment is represented by a set of 'branches' growing out from a single point. For each branch, the outcome is written at the end and the probability is written along the line. To find the probability of a specific path, you multiply the probabilities along those branches.

Equally Likely Outcomes: In many theoretical models, we assume outcomes are 'fair'. This means no outcome is weighted more heavily than another. If a spinner is divided into four equal-sized colored quadrants (red, blue, yellow, green), the visual symmetry ensures each color has a probability of $\frac{1}{4}$.