How to use this probability calculator
Enter P(A) and P(B) using decimals from 0 to 1 or percentages from 0 to 100. Choose independent, mutually exclusive, known overlap P(A ∩ B), known conditional probability P(B | A), or relationship unknown. Changing scale loads example values. Required blanks and invalid values stop the calculation. In known-conditional mode the supplied direction is specifically B given A. Conditional outputs with a zero-probability conditioning event are labeled undefined instead of being displayed as zero.
How the calculation works
Marginal probabilities describe events separately. Knowing P(A) = 0.6 and P(B) = 0.5 does not reveal how often they occur together. For example, A could describe a hypothetical student's participation in one activity and B participation in another. The events may overlap, exclude one another or have some other relationship. Their numerical probabilities alone do not identify that relationship. A useful calculation starts with a coherent sample space and an explicit statement of what A and B mean.
With an unknown relationship, the overlap must lie between max(0, P(A) + P(B) − 1) and min(P(A), P(B)). For 0.6 and 0.5, the interval is 0.1 to 0.5. The union consequently lies between 0.6 and 1. These bounds describe mathematically possible arrangements; the calculator does not estimate where within the range a real overlap lies. Selecting independence adds an assumption and narrows the result, but that assumption needs evidence outside the arithmetic.
If A and B are independent, their joint probability is P(A ∩ B) = P(A)P(B). For hypothetical independent events with probabilities 0.6 and 0.5, both occur with probability 0.3, or 30%. Their union is 0.8, neither is 0.2, and exactly one is 0.5. Independence means information about one event does not change the probability of the other when the relevant conditional probability is defined. It is not a synonym for events being different or taking place at different times.
Mutually exclusive events cannot occur together, so their overlap is zero. If their probabilities are 0.2 and 0.3, their union is 0.5 and neither is 0.5. Choosing mutually exclusive for P(A) = 0.6 and P(B) = 0.5 is inconsistent because their sum exceeds one. Two positive-probability independent events have positive overlap, so they cannot also be mutually exclusive. The calculator checks these constraints instead of silently capping an impossible union at 100%.
When the overlap is known directly, enter P(A ∩ B). For P(A) = 0.6, P(B) = 0.5 and overlap 0.2, the union is 0.9, neither is 0.1, A only is 0.4 and B only is 0.3. Exactly one is 0.7. The four disjoint regions, A only, B only, both and neither, sum to one. This provides a useful independent check and makes clear why adding the two marginals alone double-counts the shared region.
Conditional probability changes the reference event. Known-conditional mode uses P(B | A), then computes overlap as P(A)P(B | A). For P(A) = 0.6, P(B) = 0.5 and P(B | A) = 0.4, the overlap is 0.24. Dividing that overlap by P(B) gives P(A | B) = 0.48. Reversing the conditioning direction therefore changes the answer. A value labeled A given B cannot be pasted into the B given A field without first establishing the appropriate conversion.
A conditional probability is undefined when its conditioning event has probability zero under this elementary event formula. If P(B) = 0, dividing the overlap by P(B) is not meaningful, even though the overlap must also be zero. The interface labels P(A | B) as undefined in that case. Likewise, known-conditional input P(B | A) is rejected when P(A) = 0; use known-overlap mode to describe the zero overlap instead. These limits prevent undefined ratios from becoming apparently valid percentages.
Every supplied joint probability must satisfy the overlap bounds implied by the marginals. Conditional inputs must produce a joint probability within those same bounds. For example, P(A) = 0.8 and P(B) = 0.2 cannot have overlap 0.5 because the overlap would exceed event B itself. The consistency check allows only machine-rounding residuals at valid boundaries. It does not repair materially inconsistent data or estimate corrected probabilities from conflicting information. Check event definitions and denominators when an error appears.
Formulas
General addition rule
Explicit independence
Conditional relationship
Possible overlap
Worked examples
Independent events
With P(A) = 0.6 and P(B) = 0.5 under an explicit independence assumption, both = 0.3, at least one = 0.8, neither = 0.2 and exactly one = 0.5. Enter 60 and 50 only when the percentage scale is selected.
A supplied conditional probability
For P(A) = 0.6, P(B) = 0.5 and P(B | A) = 0.4, both = 0.24 and P(A | B) = 0.48. The two conditional directions have different denominators and need not be equal.
Common uses
Check two-event arithmetic
Compare inclusive OR, AND, neither and exactly-one probabilities after establishing whether the events overlap or are independent.
Inspect conditional direction
Distinguish P(B | A) from P(A | B), retain the conditioning denominator and recognize when a zero-probability condition makes a ratio undefined.
Explore unknown relationships
Use overlap and union bounds to understand what two marginal probabilities can establish without adding an unsupported independence assumption.
Accuracy and limitations
This educational calculator checks the internal consistency of two-event probability inputs. It cannot validate observed data, establish independence or supply missing empirical probabilities.
Machine-rounding residuals are clamped only after consistency checks. Inputs and outputs are ordinary floating-point numbers, so very small probabilities and displayed percentages are not arbitrary-precision or exact-rational results. This is not a prediction, risk assessment or inferential statistics model.
Sources and methodology
These primary references explain the numerical methods and units used here. Review the calculation assumptions and limitations alongside the result.
Frequently asked questions
Does the calculator assume independence?
No. The default unknown-relationship mode reports ranges. Choose independence only when your model justifies it.
Does OR include both events?
Yes. OR is the inclusive union. Use exactly one when the shared overlap must be excluded.
Are mutually exclusive events independent?
Not when both have positive probability. Exclusivity means zero overlap; independence would give the positive product of their probabilities.
What if the conditioning event has probability zero?
That conditional probability is undefined under the elementary ratio formula, not zero. The result is labeled accordingly.
Can this infer probability from observed counts?
No. Enter probabilities already defined for the same sample space. The tool does not estimate sampling uncertainty or validate a real-world model.
Does it store the probabilities I enter?
No. Calculations run in your browser without sending your probability inputs to NumUtility.