Probability Calculator

About the Probability Calculator

The Probability Calculator helps you compute basic event probabilities. Enter the probability of events A and B, then review the probability of both (A AND B), either (A OR B), not A, not B, and conditional probabilities.

Probability is the mathematical framework for quantifying uncertainty. From weather forecasts to medical tests to card games, probability theory provides the tools to reason about uncertain outcomes.

This calculator handles independent events (AND = P(A) × P(B)) and the addition rule (OR = P(A) + P(B) − P(A AND B)), plus complements. Enter probabilities as values between 0 and 1.

When This Page Helps

Combining probabilities correctly requires knowing the right formula (AND, OR, NOT). This calculator applies the rules automatically and shows results for all common combinations.

How to Use the Inputs

1. Enter the probability of event A (0 to 1, e.g. 0.3 for 30%).
2. Enter the probability of event B (0 to 1).
3. View P(A AND B), P(A OR B), P(NOT A), P(NOT B).
4. Results assume events are independent unless stated otherwise.
5. Use for risk assessment, game odds, or academic exercises.

Example Calculation

Tips & Best Practices

• Probabilities must be between 0 and 1 (or 0% and 100%).
• For independent events, P(A AND B) = P(A) × P(B).
• The complement rule: P(NOT A) = 1 − P(A).
• "At least one" problems often use complement: 1 − P(none).
• P(A OR B) counts overlap, so subtract P(A AND B) to avoid double-counting.
• For dependent events, P(A AND B) = P(A) × P(B|A).

Probability in Everyday Life

Weather forecasts, sports odds, insurance premiums, and medical test accuracy all rely on probability. Understanding basic probability helps you evaluate risks and make better decisions.

Common Misconceptions

The gambler's fallacy is the belief that past results affect independent future events. A coin has a 50% chance of heads regardless of the previous 10 flips.

Probability Theory Foundations

Modern probability theory, formalized by Kolmogorov in 1933, defines probability as a measure satisfying three axioms: non-negativity, normalization (total = 1), and countable additivity for mutually exclusive events.

Professionals in data science, engineering, and finance apply these calculations daily to model complex systems and test analytical hypotheses.

Frequently Asked Questions

• What is probability?

  Probability is a number between 0 and 1 that represents how likely an event is to occur. 0 means impossible, 1 means certain, and 0.5 means equally likely as not.

• What are independent events?

  Two events are independent if the occurrence of one does not affect the probability of the other. Flipping a coin twice produces independent events.

• What is the addition rule?

  P(A OR B) = P(A) + P(B) − P(A AND B). The subtraction prevents double-counting the overlap where both events occur.

• What is conditional probability?

  P(A|B) is the probability of A given that B has occurred. For independent events, P(A|B) = P(A). For dependent events, P(A|B) = P(A AND B) / P(B).

• How do I calculate "at least one" probability?

  Use the complement: P(at least one) = 1 − P(none). For example, P(at least one head in 3 flips) = 1 − (0.5)³ = 0.875.

• What is Bayes' theorem?

  Bayes' theorem computes P(A|B) from P(B|A): P(A|B) = P(B|A) × P(A) / P(B). It is fundamental in medical testing, spam filtering, and machine learning.