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

Worked Examples: All Three Modes

Combination — picking a committee. How many ways can you choose 2 people out of 5 for a committee, where the order of selection does not matter? Enter n = 5 and r = 2 in combination mode: C(5,2) = 5! ÷ (2! × 3!) = 120 ÷ (2 × 6) = 10 distinct committees.

Permutation — awarding medals. How many ways can gold and silver be awarded among 5 runners? Now order matters, so use permutation mode with the same inputs: P(5,2) = 5! ÷ 3! = 120 ÷ 6 = 20 possible podium outcomes — exactly twice the combination count, because each pair of runners can finish in two orders.

Basic probability — drawing a card. The chance of drawing a heart from a standard 52-card deck: enter n = 52 total outcomes and r = 13 favorable outcomes to get 13 ÷ 52 = 0.25, or 25%.

Frequently Asked Questions

What is the difference between a combination and a permutation?

Both count ways to select r items from n, but permutations treat different orders as different outcomes while combinations do not. Choosing 2 committee members from 5 is a combination (10 ways); awarding gold and silver among 5 runners is a permutation (20 ways).

Can r be larger than n?

No. You cannot select more items than exist in the set, so the calculator requires r to be between 0 and n. C(n, n) and C(n, 0) both equal 1, since there is exactly one way to take everything or nothing.

Why do I get 1 when r equals 0?

By definition 0! = 1, so C(n, 0) = n! / (0! x n!) = 1. Conceptually, there is exactly one way to choose nothing: the empty selection.

Is there a limit on how large n can be?

Factorials grow extremely fast: 170! is the largest factorial a standard JavaScript number can hold, and results lose precision well before that. For large n, exact combinatorics requires big-integer arithmetic or logarithmic methods.

How do I convert the probability result to a percentage?

Multiply by 100. A result of 0.25 means 25%. Probabilities always fall between 0 (impossible) and 1 (certain), so percentages fall between 0% and 100%.