Classical or Mathematical Probability
Introduction to Probability
Welcome to PostNetwork Academy! This article explains the fundamentals of Classical or Mathematical Probability, including definitions, examples, key characteristics, and limitations.
What You Will Learn
- The definition of Classical Probability and its core formula.
- Key properties and assumptions of Classical Probability.
- Examples: Tossing a coin and rolling a die.
- Limitations of Classical Probability: Biased coin and infinite sample space.
- A summary and recap of the concept.
What is Classical Probability?
Definition: If an experiment has \( n \) mutually exclusive, equally likely, and exhaustive cases, and \( m \) of these cases are favorable for an event \( A \), then the probability of \( A \) is given by:
\[
P(A) = \frac{\text{Number of favorable cases (m)}}{\text{Total number of cases (n)}}
\]
Properties:
- Probability values are always between \( 0 \) and \( 1 \): \( 0 \leq P(A) \leq 1 \).
- \( P(A) + P(\text{not } A) = 1 \).
Examples of Classical Probability
Example 1: Tossing a Coin
- Sample Space: \( S = \{H, T\}, \quad n(S) = 2 \)
- Event \( A \): Getting a Head \( \{H\} \), \( m = 1 \)
- Probability:
\[
P(A) = \frac{1}{2}
\]
Example 2: Rolling a Die
- Sample Space: \( S = \{1, 2, 3, 4, 5, 6\}, \quad n(S) = 6 \)
- Event \( B \): Getting an even number \( \{2, 4, 6\} \), \( m = 3 \)
- Probability:
\[
P(B) = \frac{3}{6} = \frac{1}{2}
\]
Key Characteristics of Classical Probability
Key Characteristics:
- Outcomes must be mutually exclusive.
- All outcomes must be equally likely.
- The sample space must be exhaustive.
Limitations:
- Does not work if outcomes are not equally likely.
- Cannot handle cases with infinitely large sample spaces.
Limitations of Classical Probability
Example: Biased Coin
If a coin is biased, \( P(H) \neq P(T) \), the classical definition fails because the outcomes are not equally likely.
Example: Infinite Sample Space
Probability of selecting a specific integer from \( \{-\infty, …, -2, -1, 0, 1, 2, \infty\} \):
\[
P = \frac{1}{\infty} = 0
\]
This contradicts the actual chances of occurrence.
Summary
Key Takeaways:
- Classical probability provides a simple framework based on symmetry.
- It is limited to finite, equally likely, and mutually exclusive outcomes.
- The framework forms the foundation for more advanced probability theories.
Formula Recap:
\[
P(A) = \frac{m}{n}
\]
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