Point Probability
The exact probability of exactly k successes. P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
Probability
About Binomial Distribution Calculator
Exact probabilities, tails, range checks, a distribution chart, CSV export, and binomial-test p-values in one worksheet.
P(X = k)P(X <= k)P(X >= k)RangeA/B test
About this calculator
What this does
Computes exact binomial probabilities for a given number of trials and success probability, including point probabilities, cumulative lower-tail and upper-tail values, and inclusive range probabilities. It also provides a continuity-corrected normal approximation and an exact binomial test for A/B testing scenarios.
Who it is for
Students learning probability and statistics, data analysts running A/B tests, and quality control teams evaluating defect rates against acceptance thresholds.
How it works
Enter the number of trials n, success probability p, and the target number of successes k. The calculator evaluates the binomial probability mass function for each k and sums individual probabilities for cumulative and range queries. For the A/B test mode, it treats observed successes as a binomial count under the hypothesized rate and returns exact one-sided and two-sided p-values.
Limitations
Assumes independent trials with a constant success probability, which may not hold in real-world sampling without replacement. The normal approximation is only reliable when np and n(1-p) are both at least 5.
Formula
Cumulative Lower-Tail
The probability of k or fewer successes. P(X ≤ k) = Σ P(X = i) for i = 0 to k
Cumulative Upper-Tail
The probability of k or more successes. P(X ≥ k) = 1 - P(X ≤ k-1)
Normal Approximation
Approximates binomial probabilities using the normal distribution with continuity correction. z = (k ± 0.5 - np) / sqrt(np(1-p))
How it works
Step 1
Select calculator mode
Choose Basic for general binomial probabilities or A/B Test for hypothesis testing against a hypothesized rate.
Step 2
Enter trial parameters
Type the number of trials n, success probability p, and the target success count k. Use presets like Coin, Dice, or Quality Control for quick starts.
Step 3
Choose probability type
Select P(X = k) for an exact point, P(X ≤ k) for cumulative, P(X ≥ k) for tail, or enable Range for an inclusive interval like P(3 ≤ X ≤ 7).
Step 4
Toggle normal approximation
Turn on the normal approximation to compare exact binomial results with the continuity-corrected normal estimate side by side.
Step 5
Review the distribution chart
The probability mass chart visualizes the full distribution. Hover or tap bars to see individual probabilities.
Step 6
Export the distribution table
Download the full probability table as CSV for further analysis in spreadsheet or statistical software.
Reference ranges
Success Probability p
Common values: 0.5 for fair coin flips, 1/6 ≈ 0.167 for fair dice, 0.02–0.05 for quality control defect rates, and 0.01–0.10 for typical A/B test conversion rates.
Sample Size n
Small n (10–30) for classroom examples, moderate n (50–200) for quality control, and large n (500–5000) for A/B tests and survey analysis.
Normal Approximation Rule
The approximation is generally safe when np ≥ 5 and n(1-p) ≥ 5. The closer p is to 0.5, the smaller n can be for a reliable approximation.
P-Value Thresholds
Common significance levels: 0.05 for standard hypothesis tests, 0.01 for stricter tests, and 0.10 for exploratory analysis. Lower p-values indicate stronger evidence against the null hypothesis.