Statistics and Probability - Theoretical and Experimental Probability

Probability Scale: Probability is a measure of how likely an event is to occur, ranging from $0$ to $1$. Visually, this is represented as a horizontal number line where $0$ signifies an 'Impossible' event, $0.5$ represents an 'Even Chance', and $1$ signifies a 'Certain' event. Values between $0$ and $0.5$ are 'Unlikely', while values between $0.5$ and $1$ are 'Likely'.

Theoretical Probability: This is the calculated likelihood of an event occurring based on the possible outcomes in a perfect, mathematical world. For example, if you look at a fair six-sided die, you can determine the probability of rolling a $4$ is $\frac{1}{6}$ without actually rolling the die, because you know there is one '4' out of six equal faces.

Experimental Probability (Relative Frequency): Unlike theoretical probability, this is based on actual trials or observations. If you were to flip a coin $50$ times and record the results in a tally chart, the experimental probability is the ratio of the number of times an outcome occurs to the total number of trials performed.

Sample Space: The sample space is the set of all possible outcomes for an experiment. This can be visualized using a Sample Space Grid (a table where rows and columns represent different independent events, like two dice) or a Tree Diagram, where each 'branch' represents a possible outcome, helping to count the total number of paths to a result.

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

Law of Large Numbers: This principle states that as the number of trials in an experiment increases, the experimental probability will get closer and closer to the theoretical probability. If you were to plot experimental results on a line graph over hundreds of trials, the line would eventually flatten out and align with the theoretical probability value.

Expected Frequency: This is the number of times we expect an event to happen over a specific number of trials based on its probability. For instance, if the probability of winning a game is $0.2$, and you play $100$ times, you can visualize a bar chart where the expected height for the 'wins' category would be at the $20$ mark ($100 \times 0.2$).