Can the mean be fractional?

Yes; an expected count is an average, although observed counts are integers.

Is a single trial a special case?

Yes; n = 1 gives a Bernoulli distribution.

AI Glossary / Statistics and probability / Advanced

Binomial distribution

Definition

The binomial distribution describes the number of successes in a fixed number of independent trials when each trial has two possible outcomes and the same success probability. It is written X ~ Binomial(n,p), where n is the number of trials and p is the probability of success on each trial. Success is simply the outcome being counted; it need not be desirable.

What it means

A binomial random variable is a count, such as the number of simulated checks that produce a specified result out of ten checks. It does not describe the order of those results or the waiting time until the next one occurs.
The distribution separates an individual trial's probability from the variability of a whole batch. Even when p is fixed, repeated batches usually contain different counts. The expected count is a long-run average under the model, not a promise about any particular batch.

How it works

For k successes in n trials, the probability is P(X=k) = C(n,k) p^k (1-p)^(n-k), where C(n,k) = n!/[k!(n-k)!]. The powers give the probability of one particular arrangement, and the combination factor counts how many arrangements contain k successes.
The mean is np and the variance is np(1-p). The standard deviation is the square root of that variance. Probabilities for a range of counts are found by adding the relevant point probabilities; complements often simplify calculations such as at least one success.
The assumptions need to match the process. Dependence between trials, changing probabilities or a stopping rule that does not fix the trial count can require a different model. When p is estimated rather than known, the binomial calculation conditional on that estimate does not automatically include uncertainty about p.

Three practical examples

Simulation checks: a researcher calculates the chance of a specified number of outcomes across independently repeated runs with a common probability model.
Response planning: an analyst uses a binomial model as a starting approximation for a fixed number of independent yes-or-no responses, then checks whether audience differences undermine the common-probability assumption.
Test design: a team compares the chance of seeing at least one event under different fixed trial counts, distinguishing a model-based calculation from a guarantee of discovery.

Think of it this way

Imagine a fixed tray with one slot for each trial. Every slot receives either a success marker or a failure marker. The binomial distribution describes how many success markers the completed tray contains, assuming independent placements with the same probability.

Worked example

Consider an original simulation with ten independent trials. Each has success probability p = 0.2, and X counts successes. The mean is 10 times 0.2 = 2. The variance is 10 times 0.2 times 0.8 = 1.6, giving standard deviation approximately 1.2649 successes.
The probability of exactly two successes is C(10,2) times 0.2 squared times 0.8 to the eighth power. There are 45 arrangements, so the probability is 45 times 0.04 times 0.16777216 = 0.301989888, or about 30.20 percent.
One particular arrangement, such as successes on the first two trials and failures on the remaining eight, has probability 0.04 times 0.16777216 = 0.0067108864. Multiplying by 45 accounts for every arrangement with two successes. Omitting that factor would answer a different question.
The probability of zero successes is 0.8 to the tenth power = 0.1073741824. Therefore, the probability of at least one success is 1 - 0.1073741824 = 0.8926258176, about 89.26 percent. A mean count of two clearly does not rule out a batch with no successes.
For at least three successes, subtract the probabilities of zero, one and two from one. The one-success probability is 10 times 0.2 times 0.8 to the ninth power = 0.268435456. The resulting probability is 1 - 0.1073741824 - 0.268435456 - 0.301989888 = 0.3222004736, about 32.22 percent.
Now change the process while keeping the same individual success probability. Suppose a single Bernoulli draw determines all ten outcomes together: if it succeeds, all ten succeed; otherwise all ten fail. Each trial still has marginal probability 0.2, and the mean total remains two. But the total is now either zero or ten, with probabilities 0.8 and 0.2. Its variance is 16, not 1.6, and the chance of at least one success is only 0.2.
This dependent comparison is not binomial. It demonstrates why matching n and the marginal probability is insufficient without checking dependence.

Illustrative case study

Fictional company Willow Testing runs a fixed batch of simulated checks. Its analysts initially consider a binomial count model and record the assumptions behind it. They inspect whether runs use independent randomness and whether all runs have the same configured success probability.
When checks share a common failure condition, the team investigates the resulting dependence rather than simply increasing the trial count and applying the original formula. It reports both the predicted count distribution and the assumptions that support it. This fictional workflow reports no measured reliability result.

Common misconceptions and mistakes

Success does not necessarily mean a favorable result; it might denote a fault or another event of interest. The expected count need not be the count observed, and it need not be an integer in other parameter settings.
Another mistake is assuming that any collection of binary outcomes gives a binomial total. Independent trials with unequal probabilities have a different count distribution, and dependent trials can behave very differently. Sampling without replacement from a small finite population also changes the probability after each draw. Finally, exactly k, at least k and more than k are different events and require different sums.

Business use

The binomial distribution provides a transparent baseline for counts across a fixed set of comparable independent opportunities. Reports should specify what counts as success, the trial count and how p was obtained. They should also distinguish uncertainty in the count from uncertainty in the estimated probability.

Check your understanding

Why multiply by 45 for exactly two successes?

Answer: There are 45 possible placements of those successes.

Why was at least one calculated with a complement?

Answer: Its opposite was the single event of zero successes.

Why did the dependent example fail the binomial assumptions?

Answer: All trial outcomes were determined together.

Related terms

Probability mass function (pmf)
Independence
Hypergeometric distribution
Expected value
Binomial regression
Bernoulli distribution
Probability mass function
Poisson-binomial distribution

Learn next

Compute the full probability table for the example and verify that the probabilities sum to one and the weighted count averages to two.

Sources

NIST/SEMATECH e-Handbook of Statistical Methods, Binomial Distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm . The ten-trial calculations, dependent comparison and company scenario are original illustrations.

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