Binomial Distribution Calculator
Counting Successes Across a Fixed Number of Trials
Any process with a fixed number of independent yes/no trials and a constant success probability — defective parts on a production line, free throws attempted, patients responding to a treatment — is a candidate for the binomial distribution. It answers precisely how likely a given number of successes is, not just the average you'd expect over the long run.
The Formula
Mean: μ = np
Variance: σ² = np(1−p)
n is the number of trials, k is the number of successes, and p is the probability of success on any single trial. C(n, k) is the number of distinct ways to arrange k successes among n trials.
Worked Example: Defect Rate Inspection
A production line has a historical defect rate of 5%. Inspecting 20 units at random:
| Defective units (k) | Probability |
|---|---|
| 0 | 35.85% |
| 1 | 37.74% |
| 2 | 18.87% |
| 3 | 5.96% |
Expected defects (mean) = 20 × 0.05 = 1.0, with a standard deviation of 0.97. The probability of finding 1 or fewer defective units is 73.58% — useful context when deciding whether an inspection sample of 2+ defects should trigger a process review.
Where This Matters
- Quality control — acceptance sampling plans are built directly on binomial probabilities.
- Clinical trials — modeling the number of patients who respond to treatment out of a fixed cohort.
- A/B testing — conversions out of a fixed number of visitors follow a binomial distribution.
- Polling — the number of respondents favoring a candidate out of n surveyed, assuming independent responses.
How to Use This Calculator
- Choose Exactly, At Most, or At Least k Successes from the dropdown.
- Enter the number of trials (n).
- Enter the number of successes (k) you're interested in.
- Enter the probability of success on a single trial (as a decimal between 0 and 1).
- Select Calculate to get the probability, along with the mean and standard deviation of the distribution.
Related Calculations
For the special two-outcome case of a coin, see the Coin Flip Calculator. For counting rare events over a continuous interval instead of fixed trials, use the Poisson Distribution Calculator.