Probability Calculator
Calculate combinations, permutations, and probabilities with precision. Essential for statistics, data analysis, and real-world probability scenarios.
Understanding Probability Concepts
Combinations: Number of ways to select items when order doesn't matter
Permutations: Number of ways to arrange items when order matters
Basic Probability: Likelihood of an event occurring
Key Formulas
Combination Formula: C(n,r) = n! / (r! × (n-r)!)
Permutation Formula: P(n,r) = n! / (n-r)!
Basic Probability: P(event) = favorable outcomes / total outcomes
Axiomatic Foundation of Probability
The mathematical theory of probability rests on the Kolmogorov axioms, which provide a rigorous foundation for probability calculations. These axioms establish probability as a measure on a σ-algebra of events, enabling the development of a comprehensive theoretical framework. This axiomatic approach unifies discrete and continuous probability distributions within a single mathematical structure.
The fundamental axioms lead to key probability properties and theorems, including the laws of total probability and conditional probability. These principles form the basis for more advanced concepts in probability theory and its applications in statistical inference and stochastic processes.
Combinatorial Probability
The enumeration of possible outcomes forms the basis for discrete probability calculations:
Combinations: C(n,r) = n! / (r!(n-r)!)
Permutations: P(n,r) = n! / (n-r)!
With Repetition: P(n;n1,...,nk) = n! / (n1!...nk!)
Properties:
- C(n,r) = C(n,n-r)
- C(n,0) = C(n,n) = 1
- P(n,n) = n!
- P(n,1) = n
Probability Measures and Spaces
The formal structure of probability theory involves specific mathematical components:
Sample Space (Ω): Set of all possible outcomes
Event Space (F): σ-algebra of measurable events
Probability Measure (P): Function P: F → [0,1]
Measure Properties:
- • Non-negativity: P(A) ≥ 0
- • Normalization: P(Ω) = 1
- • Countable Additivity: P(∪Aᵢ) = ΣP(Aᵢ)
Conditional Probability and Independence
The relationship between events is quantified through conditional probability:
P(A|B) = P(A∩B) / P(B)
Independence: P(A∩B) = P(A)P(B)
Chain Rule: P(A₁∩...∩Aₙ) = P(A₁)P(A₂|A₁)...P(Aₙ|A₁∩...∩Aₙ₋₁)
Bayes' Theorem:
P(A|B) = P(B|A)P(A) / P(B)
Probability Distributions
The mathematical description of random phenomena involves probability distributions:
Discrete Distributions:
- • PMF: P(X = x)
- • CDF: F(x) = P(X ≤ x)
- • E(X) = Σ xP(X = x)
- • Var(X) = E((X - μ)²)
Continuous Distributions:
- • PDF: f(x)
- • CDF: F(x) = ∫f(t)dt
- • E(X) = ∫ xf(x)dx
- • Var(X) = ∫(x - μ)²f(x)dx