Probability Calculator
Calculate two-event and binomial probabilities, including and, or, conditional, exactly, at-most, and at-least results with steps and charts.
Compare event relationships or model repeated independent trials.
Event probabilities
Results
Calculation breakdown
- Independent events: P(A and B) = P(A) x P(B) = 40% x 30% = 12%.
- P(A or B) = P(A) + P(B) - P(A and B) = 58%.
- P(exactly one) = P(A only) + P(B only) = 46%.
- P(neither) = 1 - P(A or B) = 42%.
How events A and B overlap
See the overlap as a concept map, then compare all four possible outcomes by their calculated share. The intersection represents both events occurring, while the non-overlapping regions represent only A or only B. The area outside both circles represents neither event and completes the sample space.
| Outcome | Probability |
|---|---|
| A only | 28% |
| Both A and B | 12% |
| B only | 18% |
| Neither | 42% |
Probability has no metric or imperial unit. Results are displayed to 2-3 decimal places and depend on the relationship or trial assumptions selected.
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What does the Probability Calculator calculate?
The Probability Calculator converts the values you enter into a structured estimate using the formulas and assumptions documented on this page. Use the result to compare scenarios, review the calculation details, and identify sensitive inputs. Check the guide and formula sections before using the estimate for an important decision.
Probability Calculator at a Glance
- Calculates intersections, unions, complements, exclusive outcomes, and conditional probabilities.
- Supports independent, mutually exclusive, and custom-overlap event relationships.
- Calculates exact, at-most, and at-least binomial probabilities.
- Shows expected successes, variance, standard deviation, and an accessible distribution chart.
- Accepts percent or decimal inputs and displays results to two or three decimal places.
A Short History of Probability
Probability theory developed from seventeenth-century questions about games of chance and grew into a foundation for statistics, science, insurance, engineering, and decision analysis. Modern formulas describe both relationships between events and repeated random trials.
What Does This Probability Calculator Do?
Use Two events to compare A and B, including their intersection, union, exclusive outcomes, complements, and conditional probabilities. Use Repeated trials for a fixed sequence of independent success-or-failure trials with one constant success probability. Review the Probability Calculator assumptions before relying on this result for decisions.
Who Should Use It?
Students can verify probability homework and inspect formulas. Teachers can compare event relationships. Analysts and everyday users can model quality checks, surveys, games, repeated attempts, and other clearly defined chance processes. Review the Probability Calculator assumptions before relying on this result for decisions.
What Problems Does It Solve?
For “What Problems Does It Solve?,” the Probability Calculator organizes the relevant inputs, applies the documented method, and shows a result that can be compared across scenarios. The points below identify the specific considerations and assumptions users should review before relying on the estimate.
- Finds the probability that both events, either event, exactly one event, or neither event occurs.
- Calculates conditional probabilities from a known intersection.
- Checks whether a custom overlap is mathematically feasible.
- Calculates exactly, at most, or at least k successes in n trials.
- Connects formulas to visual event regions and a binomial distribution.
Key Benefits
- Flexible: switch between percent and decimal probability inputs.
- Model-aware: formulas change with the selected event relationship.
- Transparent: worked steps show how each result is derived.
- Validated: probabilities, intersections, trial counts, and success counts are checked.
- Accessible: visualizations include labels and exact table alternatives.
How to Choose the Correct Probability Model
For “How to Choose the Correct Probability Model,” the Probability Calculator organizes the relevant inputs, applies the documented method, and shows a result that can be compared across scenarios. The points below identify the specific considerations and assumptions users should review before relying on the estimate.
- Independent: knowing one event occurred does not change the other event's probability.
- Mutually exclusive: A and B cannot happen together.
- Custom intersection: enter a known P(A and B) when events overlap but independence is not justified.
- Binomial: use a fixed trial count, two outcomes, independent trials, and a constant success probability.
Probability Definitions
- Intersection
- The outcomes shared by A and B.
- Union
- All outcomes in A, B, or both.
- Complement
- Every outcome outside an event.
- Conditional probability
- The probability of one event given that another occurred.
- Independent events
- Events that do not change one another's probability.
- Binomial trial
- An independent two-outcome trial with constant success probability.
When Should You Use This Calculator?
Use event calculations for overlapping categories, card or dice outcomes, survey responses, or risk comparisons. Use the binomial mode for fixed pass/fail checks, defect counts, repeated attempts, or other independent yes/no trials with constant p. Review the Probability Calculator assumptions before relying on this result for decisions.
Probability is dimensionless. Measurements used to define an event should still use consistent units.
How the Calculation Works
For “How the Calculation Works,” the Probability Calculator organizes the relevant inputs, applies the documented method, and shows a result that can be compared across scenarios. The points below identify the specific considerations and assumptions users should review before relying on the estimate.
- Select percent or decimal input.
- Choose two-event or repeated-trial mode.
- Select the relationship or enter n, k, and p.
- Calculate and inspect the result cards and worked steps.
- Use the diagram or exact table to verify the represented outcomes.
Review the probability formulas or follow the worked examples.
How to Interpret the Results
A probability near 0% is unlikely under the model assumptions, while one near 100% is likely. Neither value guarantees what will happen in one observation. For repeated trials, the expected value is a long-run average count rather than a promised result.
Common Probability Mistakes
- Multiplying P(A) and P(B) without confirming independence.
- Adding overlapping events without subtracting P(A and B).
- Assuming mutually exclusive events are independent.
- Confusing “at least k” with “more than k.” At least includes k.
- Using a binomial model when p changes between trials.
- Treating a model estimate as certainty.
Frequently Asked Questions
What is the difference between independent and mutually exclusive events?
Independent events do not change each other's probabilities and can occur together. Mutually exclusive events cannot occur together. For a useful Probability Calculator result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.
How do I calculate the probability of A or B?
Use P(A or B) = P(A) + P(B) - P(A and B). Subtracting the intersection prevents counting outcomes in both events twice. For a useful Probability Calculator result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.
When should I use the binomial calculator?
Use it for a fixed number of independent trials with two outcomes per trial and the same success probability on every trial. For a useful Probability Calculator result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.
What do at most and at least mean?
At most k includes every result from zero through k. At least k includes every result from k through the total number of trials. Use realistic inputs and review the Probability Calculator assumptions before relying on this result for decisions.
Can probabilities use metric or imperial units?
Probability is dimensionless. Enter it as a percent or decimal; any measurements used to define an event must already use consistent units. For a useful Probability Calculator result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.
Why can conditional probability be undefined?
P(A given B) divides by P(B), so it is undefined when P(B) is zero. For a useful Probability Calculator result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.
One Fun Fact About Probability
Two events with positive probability cannot be both independent and mutually exclusive. If they were mutually exclusive, their intersection would be zero; if independent, their intersection would equal P(A)P(B), which is positive.
Conclusion
Correct probability starts with the correct model. Define the outcomes, determine whether events overlap or influence one another, and confirm that repeated trials satisfy binomial assumptions before interpreting the result.
Continue with the step-by-step probability guide.