Quick guide
How to use this calculator
- Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
- Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
- Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the probability bounds calculator works
max(0,P(A)+P(B)−1)≤P(A∩B)≤min(P(A),P(B)).
Bounds show what is possible without assuming independence and can prevent unjustified multiplication of marginal probabilities.
Worked example
Probability Bounds example
If P(A)=0.7 and P(B)=0.6, their intersection can range from 0.3 to 0.6 without more dependence information.
max(0,P(A)+P(B)−1)≤P(A∩B)≤min(P(A),P(B)).
Supported inputs
Precision and limits
Model and design
Only the two marginal probabilities are known; narrower ranges require valid additional dependence information.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
Bounds show what is possible without assuming independence and can prevent unjustified multiplication of marginal probabilities.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
Entered values and calculated results stay in this browser and are not sent to an analytics service.
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