Page 1: Fundamental Probability Formulas and Concepts

This page introduces core probability formulas and concepts essential for understanding conditional probability and related topics.

The page begins by presenting key probability laws:

Definition: P(A) + P(Ā) = 1, where Ā is the complement of event A.

Definition: P(A∪B) = P(A) + P(B) - P(A∩B), known as the addition rule of probability.

Conditional probability formulas are then introduced:

Formula: PB(A) = P(A∩B) / P(B), which defines the probability of A given B has occurred.

The concept of probability distribution is explained using a simple example:

Example: A probability distribution for a random variable X is given as: X = xi | 0 | 1 | 2 P(X=xi) | 0.1 | 0.7 | 0.2

Expected value (mean) is defined and calculated:

Formula: E(X) = Σ xi * P(X=xi)

Example: E(X) = 0 * 0.1 + 1 * 0.7 + 2 * 0.2 = 1.1

Variance and standard deviation are introduced as measures of dispersion:

Formula: V(X) = Σ(xi² * P(X=xi)) - (E(X))²

Example: V(X) = (0² * 0.1 + 1² * 0.7 + 2² * 0.2) - 1.1² = 0.29

Formula: σ(X) = √V(X) = √0.29 ≈ 0.54

The page concludes with rules for constructing and interpreting weighted probability trees:

1. The sum of probabilities from branches at the same node equals 1.
2. The probability of an event represented by a path is the product of probabilities along that path.
3. The probability of an event A is the sum of probabilities of all paths leading to A.