Binomial Distribution Calculator
Evaluate probabilities and cumulative tails for Binomial(n, p). Enter the number of trials, probability of success, and observed successes to explore the distribution.
P(X = 5)
22.70%
Probability of exactly k successes
P(X ≤ 5)
66.52%
Cumulative probability up to k
P(X ≥ 5)
56.18%
Upper tail probability
Mean: 4.80 • Variance: 2.88 • Std. Dev: 1.70
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.0022 | 0.0022 |
| 1 | 0.0174 | 0.0196 |
| 2 | 0.0639 | 0.0834 |
| 3 | 0.1419 | 0.2253 |
| 4 | 0.2128 | 0.4382 |
| 5 | 0.2270 | 0.6652 |
| 6 | 0.1766 | 0.8418 |
| 7 | 0.1009 | 0.9427 |
| 8 | 0.0420 | 0.9847 |
| 9 | 0.0125 | 0.9972 |
| 10 | 0.0025 | 0.9997 |
| 11 | 0.0003 | 1.0000 |
| 12 | 0.0000 | 1.0000 |
How to Use This Calculator
- Specify the number of trials (n) for your binomial experiment.
- Enter the probability of success on each trial (p).
- Provide the number of successes (k) you want to evaluate.
- Review the probability tables, tail probabilities, and summary statistics.
Formula Reference
P(X = k) = C(n, k) · pk · (1 − p)n − k
Mean = n · p
Variance = n · p · (1 − p)
Std. Deviation = √(n · p · (1 − p))
This distribution models the number of successes in independent Bernoulli trials. When np(1 − p) is large, the normal approximation with continuity correction provides a quick estimate.
Frequently Asked Questions
What if the probability changes between trials?
The binomial model requires a constant probability. If p varies, consider the Poisson binomial distribution which aggregates different probabilities.
When is the normal approximation valid?
Rules of thumb: n · p ≥ 10 and n · (1 − p) ≥ 10. Apply a continuity correction by evaluating ranges like P(k − 0.5 ≤ X ≤ k + 0.5).
Can I compute cumulative probabilities without summing k values?
Statistical libraries implement efficient algorithms. This calculator precomputes the PMF table and cumulative totals for you.
Is fractional n allowed?
No. The number of trials must be an integer, though expected counts (n · p) can be fractional.