Probability Calculator — Events & Odds
Calculate single-event probability, joint probability, union, conditional probability, complement, and at-least-one probability. Step-by-step formulas shown. Everything runs in your browser.
Understanding Probability
Probability quantifies uncertainty. It assigns a number between 0 (impossible) and 1 (certain) to every event. Whether you are calculating your odds of winning a lottery, assessing risk in a business decision, or solving a homework problem, probability provides the framework for reasoning about uncertain outcomes.
Core Formulas
Single event: P(A) = favorable outcomes / total outcomes
Complement: P(not A) = 1 - P(A)
Union (or): P(A or B) = P(A) + P(B) - P(A and B)
Intersection (and, independent): P(A and B) = P(A) × P(B)
Conditional: P(A|B) = P(A and B) / P(B)
At least one in n trials: P(at least 1) = 1 - (1 - p)n
Independent vs Dependent Events
Two events are independent if the occurrence of one does not change the probability of the other. Flipping a coin twice produces independent events — getting heads on the first flip does not affect the second. Events are dependent when one outcome influences the other. Drawing two cards from a deck without replacement is dependent because the first draw changes the remaining cards.
Bayes' Theorem Introduction
Bayes' theorem relates conditional probabilities: P(A|B) = P(B|A) × P(A) / P(B). It is used in medical testing (updating disease probability after a test result), spam filters (updating spam probability after seeing certain words), and machine learning (Bayesian classifiers). The conditional probability mode in this calculator gives you the building blocks for Bayesian reasoning.
Odds vs Probability
Odds and probability express the same information differently. If the probability of rain is 0.25 (25%), the odds in favor are 1:3 (1 favorable to 3 unfavorable). Odds are commonly used in gambling and sports betting. To convert: odds = p / (1 - p), and p = odds / (1 + odds).
Common Applications
- Games of chance: Calculating poker hand probabilities, dice rolls, or lottery odds.
- Risk assessment: Estimating the probability of project delays, equipment failure, or market downturns.
- Statistics: Hypothesis testing, confidence intervals, and p-values all rely on probability theory.
- Quality control: Calculating defect rates and acceptance sampling in manufacturing.
- Insurance: Actuaries use probability to price policies and estimate claim frequencies.