Binomial Probability Calculator
Calculate exactly k, at most k and at least k successes in independent trials. Explore the binomial distribution, expected value and standard deviation.
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
A binomial model needs a fixed number of independent trials with the same probability of success each time. Multiply the binomial coefficient by p^k and (1 − p)^(n − k). Sum the appropriate probabilities for at-most and at-least results.
P(X = k) = C(n,k) p^k (1 − p)^(n − k); E[X] = np
Worked example
For 10 independent fair coin tosses, exactly 3 heads has probability C(10,3)/2¹⁰ = 120/1024 = 0.1171875, or 11.71875%.
Assumptions and input details
Trials and successes must be integers. The model is not appropriate for dependent trials or changing success probabilities. For more than 60 possible counts the chart groups neighboring counts; calculations still use every count. Extremely small probabilities may underflow floating-point precision.
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.
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