Probability Calculator

Axiomatic Probability Theory & Statistical Logic

Probability quantifies the likelihood that a particular event E will occur within a sample space S, bounded between 0 (impossible event) and 1 (certain event).

Core Probability Axioms & Formulas

• Classical Probability: P(E) = (n(E)) / (n(S)) = Favorable OutcomesTotal Sample Space

• Complement Rule: P(E') = 1 - P(E)

• Odds in Favor: Odds = P(E) : P(E') = k : (n - k)

• General Addition Rule (Union): P(A B) = P(A) + P(B) - P(A B)

• Mutually Exclusive Events: P(A B) = 0 P(A B) = P(A) + P(B)

• Independent Events (Intersection): P(A B) = P(A) × P(B)

• Conditional Probability: P(A | B) = (P(A B)) / (P(B))

• Bayes' Theorem: P(A | B) = (P(B | A) × P(A)) / (P(B))

Key Probability Distributions

• Binomial Distribution: Models k successes across n independent Bernoulli trials: P = nk p^k (1-p)^n-k.

• Normal Distribution: Gaussian bell curve for continuous random variables defined by mean μ and standard deviation σ.

How to Use This Calculator

Enter your Probability Operation — choose from Single Event: Favorable Outcomes ÷ Total Outcomes, Two Independent Events: P(A) and P(B), Two Mutually Exclusive Events: P(A) or P(B), Number of Favorable Outcomes (k) (e.g. 1), Total Possible Outcomes (n) (e.g. 6), Probability P(A) [0 to 1 or %] (e.g. 0.40), Probability P(B) [0 to 1 or %] (e.g. 0.50), and Joint Probability P(A ∩ B) [if General] (e.g. 0.20). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Probability Calculator

Input variables:

probabilityMode — Probability Operation

favorableOutcomes — Number of Favorable Outcomes (k)

totalOutcomes — Total Possible Outcomes (n)

probabilityA — Probability P(A) [0 to 1 or %]

probabilityB — Probability P(B) [0 to 1 or %]

Computed outputs:

probabilityDecimal — Probability P(E)

probabilityPercentage — Probability Percentage

complementProbability — Complement P(E')

oddsInFavor — Odds in Favor

Mathematical relationships extracted from calculation logic:

unfavorable = n - k

pIntersection = pA * pB

pUnion = pA + pB - pIntersection

primaryP = Math.min(1, pA + pB)

compPct = (1 - primaryP) * 100

Standard: ASA and NIST statistical best practices

Worked Example

Rolling a 6 on a fair six-sided die: Favorable outcomes k=1, total n=6, yielding P=0.1667 (16.67%), complement P'=83.33%, and odds in favor of 1:

5.

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.