Binomial Distribution Calculator: PMF, CDF, Table, and Inverse
Compute binomial probabilities for any comparison, read the complete k table, get the bracketed inverse with both attained probabilities, and see the normal and Poisson approximation error.
At a glance
- Computes
- Exact binomial, geometric and negative binomial probabilities, plus their inverses
- You supply
- Trials n, success probability p, a comparison, and a count k
- Use when
- A fixed number of independent trials with the same success probability
- Not for
- Counting events in an interval with no fixed number of trials Poisson Distribution Calculator
P(X ≤ 6)
0.608010
60.801%
Density by counts; 21 bins of width 1 (fd rule); Binomial distribution with n = 20 and p = 0.3; P(X ≤ 6)
0.608010
| Detail | Value |
|---|---|
| Mean | 6.00000 |
| Variance | 4.20000 |
| Standard deviation | 2.04939 |
| Mode | 6 |
| Skewness | 0.195180 |
| Excess kurtosis | -0.0619048 |
Exact result and approximations
| Method | Value | Absolute error | Relative error |
|---|---|---|---|
| Exact | 0.608010 | reference | reference |
| Normal with continuity correction | 0.5963749160 | 0.01163 | 1.914% |
| Normal without correction | 0.5000000000 | 0.1080 | 17.76% |
| Poisson (lambda = 6) | 0.6063027824 | 0.001707 | 0.2808% |
Approximation thresholds below are named conventions. Their observed error is the result that matters.
- np ≥ 10 and n(1 - p) ≥ 10
- np = 6.00000, n(1 - p) = 14.0000; not met
- np(1 - p) > 5
- np(1 - p) = 4.20000; not met
- n ≥ 100 and np ≤ 10
- n = 20.0000, np = 6.00000; not met
Bracketed inverse
Requested probabilities within about 1e-11 of a value the distribution actually attains are treated as that value, because the cumulative probability itself is only accurate to about 1e-13.
| Detail | Value |
|---|
Count table
Table cells show 6 significant figures. The readout and CSV export carry the full precision the method supports. The complete finite support is shown.
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) | In region |
|---|---|---|---|---|
| 0 | 0.000797923 | 0.000797923 | 1 | yes |
| 1 | 0.00683934 | 0.00763726 | 0.999202 | yes |
| 2 | 0.0278459 | 0.0354831 | 0.992363 | yes |
| 3 | 0.0716037 | 0.107087 | 0.964517 | yes |
| 4 | 0.130421 | 0.237508 | 0.892913 | yes |
| 5 | 0.178863 | 0.416371 | 0.762492 | yes |
| 6 | 0.191639 | 0.608010 | 0.583629 | yes |
| 7 | 0.164262 | 0.772272 | 0.391990 | |
| 8 | 0.114397 | 0.886669 | 0.227728 | |
| 9 | 0.0653696 | 0.952038 | 0.113331 | |
| 10 | 0.0308171 | 0.982855 | 0.0479619 | |
| 11 | 0.0120067 | 0.994862 | 0.0171448 | |
| 12 | 0.00385928 | 0.998721 | 0.00513816 | |
| 13 | 0.00101783 | 0.999739 | 0.00127888 | |
| 14 | 0.000218107 | 0.999957 | 0.000261047 | |
| 15 | 0.0000373898 | 0.999994 | 0.0000429400 | |
| 16 | 0.00000500756 | 0.999999 | 0.00000555025 | |
| 17 | 5.04964e-7 | > 0.999999999 | 5.42695e-7 | |
| 18 | 3.60688e-8 | > 0.999999999 | 3.77309e-8 | |
| 19 | 1.62717e-9 | > 0.999999999 | 1.66203e-9 | |
| 20 | 3.48678e-11 | 1 | 3.48678e-11 |
Equivalent calls
R pbinom(6, 20, 0.3) Python scipy.stats.binom.cdf(6, 20, 0.3) Excel =BINOM.DIST(6, 20, 0.3, TRUE) TI-84 binomcdf(20, 0.3, 6)
Fixed trials, exact counts
This page counts successes in a fixed number of independent trials that share one probability. The exact mass, cumulative probability, survival tail and quantile come from the discrete model. When there is only an average rate and no fixed number of trials, use the Poisson distribution page. For a continuous quantity, use the normal distribution page.
Geometric and negative binomial are modes here, not separate URLs. The controls name whether the count means trials, failures, or a mean and dispersion, so an equivalent R or SciPy call uses the same convention as the displayed answer.
P(X = k) = C(n,k) p^k (1-p)^(n-k)
How?
How this is calculated
Masses are evaluated in log space. Cumulative and survival probabilities use direct incomplete-beta tails, and outside ranges add the two tails directly. Every two-proportion procedure, including an A/B test, starts from binomial counts. The outside region is also the count-side idea behind Fisher's exact test.
Formula: P(X = k) = C(n,k) p^k (1-p)^(n-k)
Why the approximation rules are conventions
Textbooks use several thresholds for normal and Poisson approximations. This tool names the rules, evaluates their actual quantities, and prints the approximation error against the exact answer. Passing a convention is guidance, not proof of a small error.
Sources
- NIST SEMATECH e-Handbook, Binomial distribution (1.3.6.6.18). NIST/SEMATECH. Retrieved .
- R documentation, The Binomial Distribution. R Core Team. Retrieved .