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Enter your values, then click Calculate result.How this calculator helps
This binomial distribution calculator evaluates success counts under a model of independent trials with the same supplied success probability. Enter a whole trial count, a decimal probability from zero to one and a whole success count of interest. It reports the probability of exactly that count, at most that count and at least that count, plus the model mean and standard deviation. The probability assumption comes from you; the tool does not infer it from business results or observed data. Sampling without replacement or trials with changing probabilities need a different model rather than an undisclosed adjustment here.
How to use it
- 1
Establish independent equal-probability trials and choose whole n, decimal p and the whole success count k.
- 2
Enter the values in the displayed units and verify the source record or measurement basis.
- 3
Click Calculate result to submit the current inputs.
- 4
Read the labeled outputs with the methodology and checks. Input edits retain the previous result until Calculate is clicked again.
Formula and methodology
P(X=k) = C(n,k)p^k(1−p)^(n−k). At-most k sums counts 0 through k; at-least k sums k through n. Mean = np. Standard deviation = sqrt(np(1−p)).
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
For 10 independent trials, p = 0.5 and k = 5: exactly five has probability 252/1,024 = 24.609375%. At most five and at least five each have probability 62.3046875%. Expected successes are 5.
How to interpret your result
For 10 independent trials, p = 0.5 and k = 5: exactly five has probability 252/1,024 = 24.609375%. At most five and at least five each have probability 62.3046875%. Expected successes are 5. Numerical binomial model for 0–500 independent identical trials. No probability estimation, dependency correction, confidence interval or exact rational tail output. Tiny displayed probabilities can round to zero. Compare the main quantity with the separately labeled intermediate values before using it in another worksheet. Retain units, input basis and any selected convention with the result so it can be reproduced.
For different inputs or formulas, use Hypergeometric Calculator; Correlation Coefficient Calculator; Percent Calculator.
Related questions this calculator covers
- binomial distribution calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| n=10, p=0.5, k=5 | exact probability 24.609375%. |
| p=0 with k=0 | exact probability 100%. |
| p=1 | all probability lies at k=n. |
Common mistakes to avoid
- Entering fifty instead of 0.5 for fifty percent.
- Calling inclusive tails complementary.
- Using independent identical trials for changing or dependent outcomes.
Check n, k and the decimal probability range. For ten fair trials and five successes, independently calculate the combination 252 divided by 1,024. Verify at-most plus at-least equals one plus exact probability within rounding. Confirm the real process assumptions separately.
Authoritative reference. Methodology reference reviewed on 4 October 2026. SolvePilot applies the explicitly displayed arithmetic and its own bounded input handling; the linked documentation does not certify an individual result. Review by Mohammad Qasim is editorial and technical, not a specialist assessment of the entered situation.What can affect the result?
Success probability is a decimal input
Enter 0.5 for fifty percent, not fifty. A probability of zero means success cannot occur under the chosen model, and one means success occurs on every trial. Values outside zero through one are rejected. The result is displayed as a percentage, which is a separate presentation choice. Check the input basis before using an imported percentage value. The page does not estimate a probability from past counts or decide whether the supplied value remains applicable to future trials.
Exactly, at most and at least include different counts
Exactly k includes only one success count. At most k includes zero through k, while at least k includes k through the trial total. Both tails include k, so adding them generally exceeds one by the exact-k probability. Do not treat them as complementary events. The complement of at most k is more than k, not at least k. Writing the event in words before reading the output prevents an apparently reasonable percentage from answering the wrong question.
Independence and common probability are assumptions
The calculation assumes the outcome of one trial does not change another trial’s probability, and that every trial has the same success probability. Repeated attempts influenced by learning, shared conditions or changing opportunities can violate those assumptions. Drawing a finite sample without replacement can also change success probability across draws. Use the separate hypergeometric tool when its population-count model matches that question. The binomial arithmetic cannot test whether the real process satisfies its prerequisites.
Whole counts and endpoint cases are explicit
Trials and the requested count must be whole numbers, with k between zero and n. Zero trials are supported: exactly zero successes is certain under that model. When p is zero or one, the distribution collapses to its corresponding endpoint, avoiding ambiguous powers in the computation. The trial limit of five hundred bounds browser work and is a technical limit, not a statistical recommendation. A mean can be fractional even though an individual observed success count cannot be.
Precision is numerical rather than exact rational output
The implementation uses logarithms for combination and probability products, reducing overflow from large factorials. It sums supported probabilities numerically and formats percentage outputs to a bounded number of decimals. Extremely small values can round to zero in the interface without being mathematically impossible. Do not use a displayed zero to claim an event has no probability unless the entered endpoint assumptions actually make it impossible. For high-precision tail work, verify with an appropriate statistical system and preserve the model assumptions.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Numerical binomial model for 0–500 independent identical trials. No probability estimation, dependency correction, confidence interval or exact rational tail output. Tiny displayed probabilities can round to zero. Results use the entered assumptions and do not verify their real-world applicability. Calculator inputs are processed locally by the browser interface. Avoid entering identifying records; save only the quantities and context necessary for your own review.
Sources and review information
Frequently asked questions
Can I enter a success percentage?＋
Convert it to a decimal first: fifty percent is 0.5. The input range is zero to one.
Are at most and at least complementary?＋
No. Both include k. Their sum equals one plus the exact-k probability, apart from numerical rounding.
What if there are zero trials?＋
Only k = 0 is valid, and zero successes then has probability one under the model.
Does this handle sampling without replacement?＋
Not as a finite-population model. A hypergeometric distribution may be appropriate for that distinct intent.
Does a displayed zero mean impossible?＋
Not necessarily. Very small numerical percentages can round to zero; endpoint assumptions determine actual impossibility.