Probability Calculator
Calculate event probability from equally likely outcomes, a known probability, independent or general two-event data, a joint and conditioning probability, or Bayes' theorem. Review exact fractions, percentages, odds, outcome regions and steps.
Last Updated: July 27, 2026
Online Event Probability Calculator
Choose the information you know. Enter probabilities as decimals, fractions, percentages or scientific notation. Every supported calculation stays exact until the displayed decimal is rounded.
Your exact probability, percentage, odds and calculation steps will appear here.
Probability details
| Result | Exact | Decimal | Percentage |
|---|---|---|---|
| Event probability | — | — | Calculate an event |
Exact fractions are reduced before display.
Four disjoint outcome regions
| Region | Exact | Decimal | Percentage |
|---|
Calculation steps
- Choose a mode, enter valid probabilities and select Calculate Probability.
Recent calculations
- Your last six results will appear here.
How to Use This Probability Calculator
- Choose the mode that matches the information in your problem.
- Enter whole-number outcome counts or probabilities from 0 through 1. You may write a probability as a decimal, fraction, percentage or scientific notation.
- Select the number of decimal places for displayed approximations. Exact fractions remain unchanged.
- Select Calculate Probability.
- Review the main result, complement, odds, detailed probability table, outcome regions and derivation steps.
Use a period as the decimal mark. Do not use commas as thousands separators. Values such as 0.25, 1/4, 25% and 2.5e-1 represent the same exact probability. The tool checks range and consistency before showing any result.
What Is Probability?
Probability measures how likely an event is within a defined model. It ranges from 0 to 1, and a value of 0.5 represents an even chance. The endpoints are the lower and upper probability bounds.
P(A) = 0 is the lower probability bound
P(A) = 1 is the upper probability bound
On an elementary finite sample space, 0 describes an impossible event and 1 describes a certain event. In continuous models, an individual possible value may have probability 0, and probability 1 may mean almost sure rather than logically certain. The percentage scale carries the same information from 0% through 100%.
A probability is meaningful only after the experiment, sample space, event and assumptions are defined. A correct formula cannot repair an incomplete event definition or an invalid independence assumption.
Six Supported Probability Modes
The calculator separates common event problems so each mode asks only for the required information.
- Favorable and total outcomes: Use this when all listed outcomes are equally likely.
- One event and complement: Convert a known event probability into its complement, percentage and odds.
- Two independent events: Find both, either, neither, exactly one and the four disjoint outcome regions.
- General two events: Use known P(A), P(B) and P(A and B) without assuming independence. The known overlap may be zero.
- Conditional from joint: Divide P(A and B) by a positive P(B) to find P(A given B) directly.
- Conditional probability and Bayes: Combine a prior probability with B's likelihood under A and not A, then find P(A given B).
This page does not calculate repeated-trial binomial distributions, permutations, combinations, statistical p-values or confidence intervals. Those tasks require different inputs and assumptions.
Probability from Favorable Outcomes
When every outcome in the sample space is equally likely, divide the favorable outcome count by the total outcome count. Rolling a fair six-sided die and getting a 4 has one favorable outcome among six possible faces.
P(roll a 4) = 1/6 ≈ 0.166667 = 16.6667%
The word equally matters. Dividing counts is not valid when outcomes have different chances. For example, counting two weather labels, rain and no rain, does not imply each has probability 1/2. Use a probability estimate supported by data or a suitable model instead.
The Complement Rule
The complement of A contains every outcome where A does not occur. Because A and not A cover the whole sample space without overlap, their probabilities add to 1.
If P(A) is 35%, then P(not A) is 65%. Complement calculations are useful for phrases such as at least one in repeated-trial problems, but this calculator intentionally does not build a binomial model. It reports the complement of the selected main result.
Two Independent Events
Events A and B are independent when learning that one occurred does not change the probability of the other. Under this assumption, multiply their probabilities to find the chance that both occur.
P(A or B) = P(A) + P(B) - P(A and B)
For P(A) = 1/2 and P(B) = 1/3, the probability of both is 1/6. The probability of A or B is 1/2 + 1/3 - 1/6 = 2/3. The four disjoint regions are both, A only, B only and neither. They must be nonnegative and sum exactly to 1.
Do not select independent mode merely because the two events sound different. Sampling without replacement, shared causes and sequential decisions often create dependence. If the intersection probability is known, use the general two-events mode instead.
General Two-Event Probability and the Addition Rule
The general addition rule subtracts the intersection because adding P(A) and P(B) counts outcomes in both events twice.
Not every three-number input is a possible probability model. The intersection must stay between the Frechet bounds shown below.
The calculator checks these bounds exactly. It then derives A only, B only, neither, P(A given B) and P(B given A). If P(A and B) equals P(A) times P(B), the events are independent. If the intersection is zero, they are mutually exclusive. Nontrivial mutually exclusive events are not independent.
Conditional Probability and Bayes' Theorem
Conditional probability restricts the sample space to outcomes where the conditioning event occurred. P(A given B) divides the joint probability by P(B), provided P(B) is greater than zero.
Use Conditional from joint when these two values are known. For example, if P(A and B) = 1/5 and P(B) = 1/2, then P(A given B) = (1/5) / (1/2) = 2/5. The joint probability cannot exceed P(B), and P(B) cannot be zero.
Bayes' theorem reverses the condition after the total-probability rule combines the A and not-A branches.
P(A given B) = [P(B given A) × P(A)] / P(B)
Suppose P(A) = 1%, P(B given A) = 95% and P(B given not A) = 5%. The joint probability P(A and B) is 0.0095. Total P(B) is 0.0095 + 0.0495 = 0.059. Therefore P(A given B) is 0.0095 / 0.059 = 19/118, or about 16.1017%. The prior is small, so the not-A branch contributes most B outcomes even though B is much more likely when A occurs.
Probability Versus Odds
Probability compares favorable weight with the whole. Odds in favor compare favorable weight with unfavorable weight. For probability p, odds in favor are p to 1 - p, while odds against reverse that ratio.
Odds against = (1 - p) : p
A probability of 1/4 equals odds of 1:3 in favor and 3:1 against. A certain event has odds 1:0 in favor. An impossible event has odds 0:1 in favor. Odds ratio is a separate association measure used with two groups and is not calculated on this page.
Exact Fractions and Display Precision
The calculator parses supported inputs as reduced rational numbers backed by arbitrary-size integers. Addition, subtraction, multiplication, division, complements, bounds and Bayes calculations remain exact. The decimal selector affects only displayed approximations.
This matters near 0 and 1. A small positive probability never appears as an unqualified zero. A probability slightly below 1 never appears as an unqualified 1. The display adds digits or uses scientific notation and a complement form when the chosen precision would hide the difference.
Inputs may contain up to 50 digits in each numeric part, and scientific exponents may range from -100 through 100. Exact intermediate fractions have a 4,000-digit combined safety limit. A controlled error appears if a supported input exceeds a limit.
Input Rules and Validation
- Probability inputs must lie from 0 through 1 inclusive, or from 0% through 100% inclusive.
- Outcome counts must be nonnegative whole numbers. Total outcomes must be positive.
- Favorable outcomes cannot exceed total outcomes.
- Use a period for decimals. A comma is not accepted as a decimal mark or thousands separator.
- Fractions require a nonzero denominator. Mixed numbers are accepted but still must satisfy the probability range.
- Expressions, units, currency symbols, NaN and infinity are rejected.
- General intersections and Bayes inputs must create four nonnegative outcome regions.
- Direct conditional mode requires 0 < P(B) and P(A and B) ≤ P(B).
- Bayes mode requires 0 < P(A) < 1 because both entered branch conditionals must have a possible conditioning event.
- Bayes mode requires total P(B) to be positive before P(A given B) is defined.
Common Probability Mistakes
- Counting outcomes that are not equally likely: favorable divided by total works only with equal chances.
- Adding overlapping events without subtraction: subtract the intersection once.
- Multiplying dependent events: the simple product rule needs independence.
- Confusing mutually exclusive with independent: two positive-probability events cannot be both.
- Reversing a conditional: P(A given B) usually differs from P(B given A).
- Ignoring the base rate: Bayes' theorem requires prior probabilities, not the conditional value alone.
- Rounding early: preserve exact or unrounded values until the final display.
How to Interpret the Result
The result describes the mathematical model supplied by your inputs. It is not evidence that the assumptions are correct. Check how probabilities were estimated, whether the sample space is complete, whether events are independent and whether conditions match the decision you need to make.
Small probabilities are not the same as impossibility, and large probabilities are not guarantees. For forecasts, risk analysis, experiments or formal research, report the event definition, data source, model assumptions and rounding rule beside the probability.
Calculation Method and Review Basis
The formulas and definitions on this page were reviewed against OpenStax Probability Theory, which covers theoretical probability, complements, independence, conditional probability and Bayes' theorem, and the NIST/SEMATECH Engineering Statistics Handbook overview of Bayes' formula and prior-to-posterior reasoning.
The calculator uses reduced rational arithmetic for supported inputs. Its formulas were checked with independent exact-number oracles, boundary cases, a simulated Blogger publish cycle and browser layouts from 280px through 1440px. These checks verify the implementation, not the quality of a real-world probability estimate or its assumptions. Method reviewed July 27, 2026.
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Frequently Asked Questions
How do I calculate probability from outcomes?
When all outcomes are equally likely, divide the number of favorable outcomes by the total number of possible outcomes.
What values may a probability have?
A valid probability ranges from 0 through 1 inclusive, which is the same as 0% through 100%.
How do I find the probability of an event not happening?
Use the complement rule: subtract the event probability from 1.
How do I calculate the probability of A and B?
For independent events, multiply P(A) by P(B). For dependent events, use a known joint probability or a suitable conditional-probability rule.
How do I calculate the probability of A or B?
Add P(A) and P(B), then subtract P(A and B) once so the overlap is not counted twice.
Are independent and mutually exclusive events the same?
No. Independent events do not change each other's probabilities. Mutually exclusive events cannot occur together. Positive-probability mutually exclusive events are dependent.
What is conditional probability?
P(A given B) is the probability of A within the restricted sample space where B occurred. It equals P(A and B) divided by P(B) when P(B) is positive.
What formula does Bayes' theorem use?
P(A given B) equals P(B given A) times P(A), divided by total P(B). Total P(B) combines the A and not-A branches.
Can I enter percentages and fractions?
Yes. The calculator accepts integers, decimals, fractions, mixed numbers, percentages and scientific notation within the valid probability range.
What are odds in favor?
Odds in favor compare the probability of success with the probability of failure. For probability p, the ratio is p to 1 minus p.
Why did the calculator reject my overlap probability?
The intersection must be no larger than either event and no smaller than max of 0 and P(A) plus P(B) minus 1. Values outside those bounds cannot define a valid two-event model.
Is a p-value the same as an event probability?
No. A p-value is a tail probability calculated under a statistical null model. This calculator handles direct event relationships and does not perform a hypothesis test.
This calculator is an educational probability tool. Verify the event definitions, equal-likelihood or independence assumptions, probability sources, model consistency and rounding requirements before relying on a result.