Probability Calculator
Calculate basic probability, complements, independent events, unions, and conditional probability using decimals, percentages, or fractions.
Probability inputs can be entered as decimals, percentages, or fractions. Examples: 0.25, 25%, or 1/4.
Results
The Probability Calculator expresses uncertainty on a scale from 0 to 1, or 0% to 100%. It can evaluate a single event and common combinations, but the correct formula depends on whether events are independent, mutually exclusive, or conditional.
Conditional probability changes the sample space
The complement is 1−P(A). Independent events satisfy P(A and B)=P(A)P(B). For any two events, P(A or B)=P(A)+P(B)−P(A and B), which prevents overlap from being counted twice.
For equally likely outcomes, P(A)=favorable outcomes/total outcomes. Complement P(not A)=1−P(A).
Enter the values in their mathematical roles
Define the event and sample space before entering values. Use decimal or percentage probabilities consistently. Select the operation that matches “and,” “or,” “not,” or a conditional statement, and do not assume independence without evidence.
The governing relationship
For equally likely outcomes, P(A)=favorable outcomes/total outcomes. Complement P(not A)=1−P(A).
A compact example
A fair six-sided die has P(rolling 4)=1/6.
Valid inputs and boundary cases
Valid probabilities lie from 0 to 1. Experimental estimates depend on data quality and sample size. Odds in favor a:b correspond to probability a/(a+b); they are not entered as the decimal ratio a/b.
Read the answer in context
Independence, mutual exclusivity and conditional information change which rule is appropriate.
An independent check
Check bounds and logic. An intersection cannot exceed either component probability, a union cannot be smaller than either event, and an event plus its complement must equal 1 within rounding.
Errors that change the answer
- Adding independent probabilities for an “and” event
- Multiplying events that are dependent
- Forgetting overlap in an “or” event
- Confusing odds with probability
- Using percentages as whole numbers
Continue with a related calculator
Continue with the Triangle Calculator or Surface Area Calculator. For everyday arithmetic and order-of-operations checks, use the Basic Calculator.
What this numerical result cannot decide
A numerical probability is only as valid as its assumptions and data. Rare-event estimates, medical risk, financial forecasts, and correlated outcomes require domain-specific models and uncertainty analysis.
A second way to understand the method
P(A|B)=P(A and B)/P(B) asks for the chance of A after B is known to have occurred. The denominator is no longer the original full sample space. If learning B changes the chance of A, the events are dependent.
Counting equally likely outcomes
When every outcome is equally likely, probability equals favorable outcomes divided by total outcomes. Drawing one ace from a standard 52-card deck has probability 4/52=1/13. This shortcut fails when outcomes have different weights, so verify the model before counting.
Sampling without replacement creates dependence
If two cards are drawn without replacement, the second probability changes after the first draw because only 51 cards remain. The chance of two aces is 4/52×3/51. With replacement, the second factor would again be 4/52. The physical procedure determines whether multiplication uses identical or conditional probabilities.
Expected counts are not guarantees
A probability of 0.2 over 100 independent trials gives an expected count of 20, but an actual run can produce more or fewer. Expected value describes a long-run average across repetitions, not a promise for one sample. Variation should be considered before planning capacity or interpreting an unusual result.
Bayes’ rule reverses a condition carefully
P(A|B) is generally not equal to P(B|A). Bayes’ rule combines the likelihood P(B|A), prior P(A), and overall evidence P(B). This distinction is crucial in screening tests, classification, and risk communication, where confusing the two conditional directions can greatly exaggerate or understate the conclusion.
Probability Calculator FAQ
Can probability be greater than 1?
No. Values outside 0–1 indicate an input or model error.
What does mutually exclusive mean?
The events cannot occur together, so their intersection is zero.
Are coin tosses always independent?
Ideal separate tosses are modeled as independent, but real experiments still rely on assumptions.
For additional worked mathematics, see the free OpenStax Prealgebra 2e reference.
