Probability Theory: Fundamental Concepts and Formulas

Fundamental Concepts of Probability

• Random Experience: An experiment whose outcome depends on chance.
• Random Event: An event that may or may not occur depending on chance.
• Sample Space (E): The set of all possible outcomes of a random experiment.
• Event: Any subset of E, including individual elementary events, the empty set, and the certain event.

Set Operations in Probability

• Union (A∪B): An event comprising elements of A or B; verified when at least one occurs.
• Intersection (A∩B): An event consisting of elements common to both A and B.
• Difference (A\B): An event consisting of elements in A that are not in B.
• Complementary Event (A'): The opposite event (E \ A).
• Mutually Exclusive Events: Two events are inconsistent if they have no common elements (A∩B = ∅).

Frequency and Axioms

• Absolute Frequency f(S): The number of times event S occurs.
• Relative Frequency Fr(S): The ratio f(S) / N.
• Axiom 1: For any event S, P(S) ≥ 0.
• Axiom 2: If two events are incompatible, the probability of their union equals the sum of their probabilities.
• Axiom 3: The total probability is 1: P(E) = 1.

Probability Theorems

• T.1: P(A') = 1 - P(A)
• T.2: P(∅) = 0
• T.3: If A ⊆ B, then P(B) = P(A) + P(B \ A)
• T.4: If A ⊆ B, then P(A) ≤ P(B)
• T.5: P(A∪B∪C...) = P(A) + P(B) + P(C) + ... (for disjoint events)
• T.6: P(A∪B) = P(A) + P(B) - P(A∩B)
• T.7: If E is finite, P(S) = Σ P(xi)

Advanced Probability Rules

• Laplace's Law: P(S) = (Number of favorable outcomes) / (Number of possible cases).
• Conditional Probability: P(A|C) = P(A∩C) / P(C).
• Independent Events: Events are independent if P(A|C) = P(A) and P(C|A) = P(C). Consequently, P(A∩C) = P(A) × P(C).
• Compound Tests: Tests involving two or more stages. They are independent if results do not influence each other, and dependent otherwise.
• Total Probability: P(S) = P(A1)P(S|A1) + P(A2)P(S|A2) + ... + P(W)P(S|W).

Rate of Change Formulas

• TVM [a, b]: (f(b) - f(a)) / (b - a)
• TVM [a, a + h]: (f(a + h) - f(a)) / h