Conditional Probability and Bayes Calculator
Update an event probability after a positive or negative signal. See the base-rate effect, a four-outcome probability table and expected counts for quality-control scenarios.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Partition the population into A and not A. Multiply each prior by its conditional positive or negative signal probability. Add the two positive paths for P(+), then divide the A-and-positive path by that total for P(A|+). Do the same for a negative signal.
P(A|+)=P(+|A)P(A) / [P(+|A)P(A)+P(+|not A)P(not A)]
Worked example
Suppose 1% of manufactured items have a defect, a check flags 90% of defective items and also flags 5% of nondefective items. In 10000 items the expected counts are 90 true flags and 495 false flags. The probability of a defect among flagged items is 90/585≈15.3846%, rather than 90%.
Assumptions and input details
A is the event of interest and + is an observed positive signal. Enter probabilities between 0 and 100%, and a population integer from 1 to 1e9. The population scales expected counts only; fractional expected counts are valid. An impossible conditioning signal has an undefined posterior. Rates are supplied by you; this arithmetic does not estimate sampling uncertainty, causal effects or independence between repeated signals.
References: Penn State Bayes theorem lesson
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Calculate a two-draw color event with or without replacement. Follow a probability tree, multiply along each path and add the matching outcomes.
Calculate exactly k, at most k and at least k successes in independent trials. Explore the binomial distribution, expected value and standard deviation.
Find the union, intersection, difference, symmetric difference or complement of two finite sets. Compare the element lists, counts and highlighted Venn regions.
Reserved advertising space.