MasterTool

Binomial Distribution Calculator

Calculate binomial distribution using your data.

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What this tool does

Calculate binomial distribution using your data. Trials must be independent, with only two outcomes and the same success probability on every trial.

How to use the Binomial Distribution Calculator

  1. Enter or select trials (n).
  2. Enter or select success probability per trial (%).
  3. Enter or select successes (k).
  4. Read the calculated result; change any measurement to compare alternatives.

Formula

P(X=k) = C(n,k) p^k (1−p)^(n−k); cumulative probabilities sum the PMF; mean = np, SD = sqrt(np(1−p))
nn
Trials (n)
pp
Success probability per trial (%)
kk
Successes (k)

Trials must be independent, with only two outcomes and the same success probability on every trial.

Worked example

For binomial distribution calculator, the following measurements illustrate the exact method: Trials (n): 10; Success probability per trial (%): 50; Successes (k): 3.

Inputs

  • Trials (n)10
  • Success probability per trial (%)50
  • Successes (k)3

Result

  • P(X = k) — exactly (%)11.72
  • P(X ≤ k) — at most (%)17.19
  • P(X ≥ k) — at least (%)94.53
  • Mean (np)5
  • Standard deviation1.58

Results explained

P(X = k) — exactly (%)
P(X = k) — exactly (%) from the formula above. Trials must be independent, with only two outcomes and the same success probability on every trial.
P(X ≤ k) — at most (%)
P(X ≤ k) — at most (%) from the formula above. Trials must be independent, with only two outcomes and the same success probability on every trial.
P(X ≥ k) — at least (%)
P(X ≥ k) — at least (%) from the formula above. Trials must be independent, with only two outcomes and the same success probability on every trial.
Mean (np)
Mean (np) from the formula above. Trials must be independent, with only two outcomes and the same success probability on every trial.
Standard deviation
Standard deviation from the formula above. Trials must be independent, with only two outcomes and the same success probability on every trial.

Frequently asked questions

P(X=k) = C(n,k) p^k (1−p)^(n−k); cumulative probabilities sum the PMF; mean = np, SD = sqrt(np(1−p))

Trials must be independent, with only two outcomes and the same success probability on every trial.

Enter numbers only, in the units and format each label describes. Remove missing values rather than substituting zero, unless zero is a real observation.

No. Results describe the numbers you entered. Statistical inference also depends on sampling design, independence, model fit and interpretation; a p-value is not the probability that a hypothesis is true.

No. Every calculation, including the distribution algorithms, runs entirely in your browser.