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A binomial experiment is a random process made of a fixed number of independent trials, where each trial has only two possible outcomes, called success and failure, and the probability of success is the same on every trial. When those conditions hold, the number of successes follows a binomial distribution.
Definition: what counts as a binomial experiment
OpenStax Introductory Statistics 2e lists three characteristics: there are a fixed number of trials; each trial has only two possible outcomes, called success and failure; and the n trials are independent and repeated under identical conditions. The random variable X is the number of successes in the n trials. The same textbook writes this as X ~ B(n, p).
Sources split or combine these conditions differently, which is why you will see three, four or five in different places (see the table below). They describe the same experiment. The plural, “binomial experiments”, simply refers to any process that passes the test.
The conditions, and why sources count three to five
| Condition | OpenStax Introductory Statistics 2e | NIST/SEMATECH e-Handbook |
|---|---|---|
| Fixed number of trials | Stated: “a fixed number of trials” | Implicit in the formula, where x runs from 0 to n |
| Two outcomes per trial | Stated: success and failure | Stated: exactly two mutually exclusive outcomes |
| Independent trials, identical conditions | Stated | Not listed as a separate condition |
| Constant probability of success | Follows from identical conditions: p is the probability of success on one trial | Stated: p is fixed for all trials |
| X counts the successes | Stated | Stated through the distribution of x |
A source that lists three conditions folds constant p into “identical conditions”; one that lists five splits out constant p and the definition of X. In practice, run a four-part check: fixed n, two outcomes, independent trials, constant p.
Binomial experiments versus other models
One trial on its own is a Bernoulli trial; see Bernoulli distribution vs binomial distribution. If you repeat trials until the first success instead of fixing n, the model changes; see binomial vs geometric.
Formula with every symbol explained
P(X = x) = C(n, x) × px × (1 − p)n − x, for x = 0, 1, …, n, where C(n, x) = n! / [x! (n − x)!] (NIST/SEMATECH e-Handbook of Statistical Methods).
- n: the number of trials, fixed in advance.
- x: the number of successes you are asking about.
- p: the probability of success on one trial; q = 1 − p is the probability of failure, so p + q = 1.
- C(n, x): the number of different orders in which x successes can occur among n trials.
- Mean = np. Standard deviation = √(np(1 − p)), so the variance is np(1 − p).
For more on the parameters and the mean, see binomial distribution parameters and the expected value of a binomial distribution.
Worked example with real numbers
OpenStax Example 4.13 reports that 41% of adult workers have a high school diploma but no further education. Suppose a random sample of 20 adult workers is drawn and X is the number in the sample with that profile.
- Check the conditions: n = 20 trials; each worker either has that profile (success) or not (failure); workers are sampled independently; p = 0.41 for each. So X ~ B(20, 0.41) and q = 0.59.
- Probability of exactly 8: P(X = 8) = C(20, 8) × 0.418 × 0.5912.
- C(20, 8) = 125,970. Also 0.418 ≈ 0.00079849 and 0.5912 ≈ 0.0017792.
- Multiply: 125,970 × 0.00079849 × 0.0017792 ≈ 0.1790.
- Mean: μ = np = 20 × 0.41 = 8.2.
- Standard deviation: σ = √(20 × 0.41 × 0.59) = √4.838 ≈ 2.20.
- Cumulative: P(X ≤ 12) ≈ 0.9738, the value OpenStax reports for this example.
Interpretation: reading the results
| Result | Value | How to read it |
|---|---|---|
| P(X = 8) | 0.179 | About 18 in 100 samples of 20 would contain exactly 8 workers with that profile. |
| Mean np | 8.2 | The long-run average number of successes per sample of 20; it need not be a whole number. |
| Standard deviation | 2.20 | Typical sample-to-sample spread of the count around 8.2. |
| P(X ≤ 12) | 0.974 | Counts of 12 or fewer cover about 97% of samples, so 13 or more would be unusual. |
Assumptions and common mistakes
Non-examples
- Ages of 100 people: many possible outcomes, not two.
- Rolling a die until a 5 appears: n is not fixed, so the count of trials is not binomial.
- A success probability that changes between trials: NIST states the binomial distribution assumes p is fixed for all trials.
Sampling without replacement
The independence condition most often fails silently when you sample from a small population without replacement. SciPy’s documentation describes the hypergeometric distribution as counting items in a sample drawn without replacement from a total population. Take 5 cards from a standard 52-card deck and count hearts (13 hearts in the deck):
| Hearts in 5 cards | Binomial B(5, 0.25) | Hypergeometric (52, 13, 5) |
|---|---|---|
| Exactly 0 | 0.2373 | 0.2215 |
| Exactly 2 | 0.2637 | 0.2743 |
The values differ because each card drawn changes the odds for the next. The hypergeometric values are the correct ones here. For that model, see the hypergeometric distribution formula.
How to report it (APA example)
Example: “Assuming a binomial model with n = 20 and p = 0.41, the probability of observing exactly 8 successes was 0.18 (M = 8.2, SD = 2.20).” State n, p, the event, and why the binomial conditions apply, and follow your style manual’s edition for number formatting.
How to calculate it in Excel, R or Python
Excel (BINOM.DIST: successes, trials, probability, cumulative):
=BINOM.DIST(8, 20, 0.41, FALSE) returns 0.1790 for P(X = 8)
=BINOM.DIST(12, 20, 0.41, TRUE) returns 0.9738 for 12 or fewerR:
dbinom(8, size = 20, prob = 0.41) # 0.1790
pbinom(12, size = 20, prob = 0.41) # 0.9738Python (SciPy takes n and p as shape parameters):
from scipy.stats import binom
binom.pmf(8, 20, 0.41) # 0.1790
binom.cdf(12, 20, 0.41) # 0.9738For a ready-made tool, see the binomial probability calculator guide.
FAQ
What are the conditions of a binomial experiment?
A fixed number of trials, two outcomes per trial (success or failure), independent trials, and the same probability of success on every trial.
What is the difference between a binomial experiment and a binomial distribution?
The experiment is the process that meets the conditions; the distribution describes the probabilities of the number of successes X it produces.
Are binomial experiments always about coins?
No. Any yes/no outcome that meets the conditions qualifies, such as whether a sampled worker has a given education profile.
Does sampling without replacement make it non-binomial?
Draws without replacement from a finite population are hypergeometric, not independent trials, as the card example shows.
Sources
- OpenStax, Introductory Statistics 2e, 4.3 Binomial Distribution: https://openstax.org/books/introductory-statistics-2e/pages/4-3-binomial-distribution (checked 2026-10-29).
- NIST/SEMATECH e-Handbook of Statistical Methods, Binomial distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm (checked 2026-10-29).
- SciPy documentation, scipy.stats.binom: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binom.html (checked 2026-10-29).
- SciPy documentation, scipy.stats.hypergeom: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.hypergeom.html (checked 2026-10-29).
