Two-Draw Probability Calculator
Calculate a two-draw color event with or without replacement. Follow a probability tree, multiply along each path and add the matching outcomes.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Assume all items are equally likely on each draw. Without replacement, subtract the first drawn item before finding the second-draw probability. Multiply conditional probabilities along a path and add the paths matching the selected event.
P(first and second)=P(first)×P(second | first); event probability=sum of matching path probabilities
Worked example
A bag with 3 red and 2 blue items gives P(two red without replacement)=(3/5)(2/4)=3/10. Exactly one red has two paths: (3/5)(2/4)+(2/5)(3/4)=3/5. With replacement, P(two red)=(3/5)²=9/25.
Assumptions and input details
Enter whole counts from 0 to 1000 for each color. Without replacement the total must be at least two; with replacement it must be at least one. Draw order matters in the four-path sample space. Red/blue paths are not generally equally likely; use their probabilities rather than just counting colored paths. Impossible first branches have no defined second conditional probability and contribute zero. Fractions are exact, percentages are rounded. Only two colors and exactly two draws are modeled.
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?
The path and event fractions use exact integer arithmetic. Percentages are rounded for display.
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.
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