Probability Calculator
Calculate the probability of events occurring. Find single event probability, probability of multiple events (AND/OR), complementary probability, and conditional probability. Click any result to copy it to your clipboard.
How to Use This Probability Calculator
This comprehensive probability calculator handles six fundamental types of probability calculations. Each calculation section is designed for specific scenarios you will encounter in statistics, games of chance, risk assessment, and decision making:
- Single Event Probability: Enter the number of favorable outcomes and total possible outcomes to find the basic probability. For example, if a bag contains 3 red balls and 7 blue balls, enter 3 favorable outcomes and 10 total outcomes to find the probability of drawing a red ball (0.30 or 30%).
- Probability of A AND B (Independent Events): When you need both events to occur and they do not affect each other, use the multiplication rule. Enter each probability as a decimal between 0 and 1. For instance, the probability of flipping heads twice in a row is 0.5 x 0.5 = 0.25.
- Probability of A OR B (Mutually Exclusive): Use this when events cannot happen simultaneously. Rolling a 2 OR a 5 on a die are mutually exclusive because you cannot roll both numbers at once. Simply add the probabilities together.
- Probability of A OR B (Non-Exclusive): When events can occur together, you must subtract the overlap to avoid counting it twice. For example, drawing a red card OR a king from a deck requires subtracting the probability of red kings.
- Complementary Probability: Calculates the probability that an event does NOT occur. This is simply 1 minus the probability of the event occurring. Often easier than calculating complex probabilities directly.
- Conditional Probability: Finds the probability of event A occurring given that event B has already happened. This is crucial in medical testing, quality control, and updated predictions based on new information.
All probabilities should be entered as decimals between 0 and 1. For example, enter 0.5 for 50% or 0.75 for 75%. Click any result to copy it to your clipboard.
What is Probability?
Probability is the mathematical study of uncertainty and randomness. It quantifies how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). A probability of 0.5 means the event has an equal chance of occurring or not occurring, like flipping a fair coin.
The foundations of probability theory were developed in the 17th century by mathematicians Blaise Pascal and Pierre de Fermat while analyzing gambling problems. Today, probability underpins virtually every field that deals with uncertainty: weather forecasting, medical diagnosis, insurance pricing, financial modeling, quality control, artificial intelligence, and scientific research.
Understanding probability helps you make better decisions under uncertainty. It teaches you to distinguish between likely and unlikely outcomes, to recognize when events are independent or related, and to update your beliefs rationally when you receive new information. These skills are valuable whether you are evaluating medical test results, assessing business risks, or simply deciding whether to bring an umbrella.
Probability Formulas and Rules
Basic Probability: P(A) = Number of favorable outcomes / Total number of possible outcomes
This fundamental formula assumes all outcomes are equally likely, such as rolling a fair die or drawing from a well-shuffled deck.
Multiplication Rule (Independent Events): P(A AND B) = P(A) x P(B)
Two events are independent when the occurrence of one does not affect the probability of the other. Each coin flip is independent of previous flips.
Addition Rule (Mutually Exclusive): P(A OR B) = P(A) + P(B)
Mutually exclusive events cannot occur simultaneously. You cannot roll a 3 and a 4 on a single die roll.
General Addition Rule: P(A OR B) = P(A) + P(B) - P(A AND B)
When events can occur together, subtract the intersection to avoid double-counting the overlap.
Complement Rule: P(NOT A) = 1 - P(A)
The probability of an event not occurring equals one minus the probability of it occurring.
Conditional Probability: P(A|B) = P(A AND B) / P(B)
The probability of A given B is the probability of both occurring divided by the probability of B. This formula is the basis for Bayes' theorem.
Understanding Odds vs Probability
While probability expresses likelihood as a fraction of 1 (or percentage of 100), odds express the ratio of favorable to unfavorable outcomes. If the probability is 0.25 (25%), the odds are 1:3, meaning one favorable outcome for every three unfavorable ones. Odds of 1:1 correspond to a probability of 0.5 (50%). Converting between odds and probability is essential when interpreting betting lines, medical statistics, or risk assessments.