Study for Statistics Test Level 2 (5)

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Today's Study: Probability

Terms:
Definition of Probability:
- Kolmogorov's Axioms:
- Specific Definitions:
  - Laplace's Definition of Probability:
  - Frequency-based Definition of Probability:
Theorems of Probability:

Terms:

Trial: An experiment or observation that results randomly.
Elementary Event: Each result of a trial.
Event: A set of elementary events.
Whole Event: The set of all elementary events.
Union: When at least one result occurs in several events. Denoted as: \[ A \cup B \]
Intersection: When each event occurs at the same time. Denoted as: \[ A \cap B \]
Empty Set or Null Set: The event where nothing occurs. Denoted as: \[ \emptyset \]
Complement: The event excluded from a particular event in the whole event. Denoted as: \[ A^c \]
Mutually Exclusive: When the intersection between two events is an empty set.

Definition of Probability:

Probability is a function defined with the following features:

Kolmogorov's Axioms:

For any event \( A \), the probability is non-negative and does not exceed 1: \[ 0 \leq P(A) \leq 1 \]
The probability of the sample space \( S \) is 1: \[ P(S) = 1 \]
For any sequence of mutually exclusive events \( A_1, A_2, \dots \): \[ P\left( \bigcup_{i=1}^{\infty} A_i \right) = \sum_{i=1}^{\infty} P(A_i) \]

Specific Definitions:

Laplace's Definition of Probability:

The probability \( P(A) \) of an event \( A \) is given by: \[ P(A) = \frac{n(A)}{n(S)} \] where \( n(A) \) is the number of favorable outcomes for event \( A \) and \( n(S) \) is the total number of possible outcomes.

Frequency-based Definition of Probability:

The probability \( P(A) \) of an event \( A \) is given by: \[ P(A) = \lim_{{n \to \infty}} \frac{m}{n} \] where \( m \) is the number of times event \( A \) occurs in \( n \) trials.

Theorems of Probability:

Addition Theorem of Probability:

The probability \( P(A \cup B) \) of the union of events \( A \) and \( B \) is given by: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] where \( P(A \cap B) \) is the probability that both events \( A \) and \( B \) occur.

Conditional Probability:

Given an event \( B \), the conditional probability of event \( A \) is: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] where \( P(B) > 0 \).

Multiplication Theorem:

The probability of the intersection of events \( A \) and \( B \) is: \[ P(A \cap B) = P(A|B) \times P(B) \] or \[ P(A \cap B) = P(B|A) \times P(A) \]

Bayes' Theorem :

The probability of event \( A_i \) given event \( B \) is: \[ P(A_i|B) = \frac{P(B|A_i) \times P(A_i)}{\sum_{j} P(B|A_j) \times P(A_j)} \] where:

\( A_i \) is a specific event from the set of possible events \( \{A_1, A_2, \dots\} \).
The summation over \( j \) represents the sum over all possible events \( A_j \).