Statistics and Probability - Theoretical and experimental probability

Theoretical Probability is based on the assumption that all outcomes in a sample space are equally likely. It is calculated by considering the possible outcomes without actually performing an experiment. On a probability scale, which is a horizontal line ranging from $0$ (impossible) to $1$ (certain), theoretical probability helps determine where an event sits based on logic.

Experimental Probability, also known as Relative Frequency, is determined through actual trials or observations. It is the ratio of the number of times an event occurs to the total number of trials performed. In a table format, you would record the 'Frequency' of each outcome and divide it by the 'Total Frequency' to find the experimental probability.

The Sample Space represents the set of all possible outcomes of an experiment. This can be visually organized using a Tree Diagram, where each branch represents a possible choice, or a Sample Space Grid (Coordinate Grid), which is particularly useful for visualizing outcomes of two independent events like rolling two dice simultaneously.

Complementary Events are outcomes that are the opposite of a specific event. The complement of event $A$ is denoted as $A'$. Visually, in a Venn Diagram where a circle represents event $A$ inside a rectangular universal set, the complement $A'$ is the entire shaded region outside the circle but within the rectangle.

The Law of Large Numbers states that as the number of trials in an experiment increases, the experimental probability (Relative Frequency) will get closer and closer to the theoretical probability. This can be seen on a line graph where the horizontal axis is the 'Number of Trials' and the vertical axis is 'Relative Frequency'; the line will fluctuate wildly at first but eventually flatten out toward the theoretical value.

Mutually Exclusive Events are events that cannot happen at the same time. For example, rolling a $2$ and a $5$ on a single die at the same time is impossible. In a Venn Diagram, these events are represented by two separate circles that do not overlap or intersect.

The Sum of Probabilities for all possible mutually exclusive outcomes in a sample space must always equal $1$. This is because it is certain that one of the outcomes in the sample space will occur. If you add up all the segments in a probability pie chart, the total area will represent a probability of $1$.