Classical Probability Model
The Classical Probability Model is one of the foundational frameworks in probability theory. It applies when all outcomes of an experiment are equally likely.
Formal Definition
If a random experiment has a finite sample space $S = \{s_1, s_2, \ldots, s_n\}$ where all $n$ outcomes are equally likely, then the probability of an event $A$ is:
$$ P(A) = \frac{|A|}{|S|} = \frac{\text{Number of outcomes favorable to } A}{\text{Total number of possible outcomes}} $$
where $|A|$ denotes the number of elements in event $A$ and $|S|$ denotes the total number of elements in the sample space $S$.
Key Terminology
- Random Experiment: A process whose outcome cannot be predicted with certainty (e.g., rolling a die, flipping a coin).
- Sample Space ($S$): The set of all possible outcomes of an experiment.
- Event ($A$): A subset of the sample space — any collection of outcomes.
- Equally Likely Outcomes: Each outcome has the same probability: $P(\{s_i\}) = \frac{1}{n}$ for all $i$.
- Favorable Outcomes: Outcomes that belong to the event of interest.
Conditions for the Classical Model
- The sample space must be finite — there are a countable, limited number of outcomes.
- All outcomes must be equally likely — no outcome is more probable than another.
- The experiment must be well-defined — outcomes are mutually exclusive and collectively exhaustive.
⚠️ Caution: The classical model does NOT apply when outcomes are not equally likely. For example, if a die is loaded (weighted), you cannot use this model directly.