What this calculator does
The binomial distribution describes every possible outcome of a fixed number of independent trials, each with the same probability of success, laid out as a full probability table from zero successes up to the maximum possible. Where a single binomial probability calculation answers "what is the chance of exactly k successes," the distribution is the whole shape at once: every k value, its individual probability, and the cumulative probability building up to it.
This calculator takes the number of trials and the probability of success on each one, and returns that entire distribution, along with the summary figures that describe its shape: the mean (expected number of successes), the variance and standard deviation (how spread out the results typically are), the mode (the single most likely outcome), and the skewness (whether the distribution leans toward fewer or more successes than the mean).
The formula
Mean is the number of trials multiplied by the probability of success (n × p). Variance is n × p × (1 − p), and standard deviation is its square root. Each individual probability P(X = k) uses the binomial probability formula, the number of ways to choose k successes from n trials, multiplied by the probability of exactly that many successes and exactly that many failures. Running that formula for every k from 0 to n, and adding a running total, builds the full distribution and its cumulative probabilities.
| Term | Meaning |
|---|---|
| n | The number of independent trials. |
| p | The probability of success on each individual trial, assumed constant across all trials. |
| P(X = k) | The probability of exactly k successes out of n trials. |
| Mode | The single value of k with the highest individual probability: the most likely outcome. |
The inputs explained
| Field | What to enter |
|---|---|
| Number of trials (n) | The number of independent trials. The full distribution table is shown for up to 60 trials. |
| Probability of success each trial (%) | The probability of success on any one trial, as a percentage, assumed the same for every trial. |
When to use it
Understanding the shape of a pass/fail process
Quality inspections, coin flips, and any process with a fixed success probability repeated many times produce outcomes following a binomial distribution; seeing every probability at once shows which outcomes are actually plausible, not just the average.
Comparing the spread of two different trial counts
The same success probability produces a tighter, more predictable distribution as the number of trials rises, since standard deviation grows more slowly than the mean; comparing the full table at two different values of n shows that directly.
Checking how skewed a small or extreme-probability distribution is
A binomial distribution is symmetric only when the success probability is exactly 50%; at low or high probabilities, especially with few trials, it leans noticeably to one side, which the skewness figure and the shape of the table both show.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the shape of the distribution change with the success probability, at 20 trials?
A fixed 20 trials, across a range of success probabilities.
| Probability of success | Mean (expected successes) | Standard deviation | Skewness |
|---|---|---|---|
| 10% | 2.000 | 1.342 | 0.5963 |
| 20% | 4.000 | 1.789 | 0.3354 |
| 30% | 6.000 | 2.049 | 0.1952 |
| 50% | 10.000 | 2.236 | 0 |
| 70% | 14.000 | 2.049 | -0.1952 |
How does standard deviation change as the number of trials grows, at 25% success probability?
A fixed 25% probability, across a range of trial counts.
| Number of trials | Mean (expected successes) | Standard deviation | Skewness |
|---|---|---|---|
| 5 | 1.250 | 0.9682 | 0.5164 |
| 10 | 2.500 | 1.369 | 0.3651 |
| 20 | 5.000 | 1.936 | 0.2582 |
| 40 | 10.000 | 2.739 | 0.1826 |
| 60 | 15.000 | 3.354 | 0.1491 |
Questions
How is this different from a plain binomial probability calculator?
A binomial probability calculator usually answers a single question: the chance of exactly k successes, or at most k, for one specified k value. This calculator instead builds the probability of every possible outcome from 0 to n at once, so the whole shape of the distribution is visible in one table rather than one point at a time.
Why is the table limited to 60 trials?
A full distribution table becomes unwieldy well before very large n, and for most practical purposes the shape is already clear by that point. For a specific single value of k at a larger n, a dedicated binomial probability calculation for just that one outcome is the more direct tool.
What does skewness of 0 actually mean here?
A skewness of 0 means the distribution is symmetric around its mean, which for the binomial distribution happens only when the success probability is exactly 50%. Any other probability produces some asymmetry, larger at more extreme probabilities and smaller trial counts.
Why can the mode differ from the mean?
The mean, n × p, is rarely a whole number and represents an average across many repetitions, while the mode is the single most probable whole-number outcome in any one set of trials. They coincide closely when p is near 50% and drift apart at more extreme probabilities.
For the probability of one specific number of successes, or "at least one," rather than the full distribution, see the binomial probability calculator. For the normal distribution that a binomial distribution approaches as trials grow large, see the normal distribution probability calculator.