Updated 12 September 2026

Binomial distribution

The binomial distribution counts successes in $n$ independent trials, each with success probability $p$. It is the backbone of coin flips, A/B tests, quality control, and opinion polls.

What to notice

• Symmetric at p = 0.5. The probabilities are symmetric whenever p = 0.5, even for small n when a normal approximation is poor. Slide p away from centre and the distribution skews toward the favoured outcome.
• The Normal approximation (dashed curve) kicks in when both $np$ and $n(1-p)$ exceed about 5. The mean is $\mu = np$, the standard deviation $\sigma = \sqrt{np(1-p)}$.
• Raising n stretches the distribution and sharpens the Normal fit — the Central Limit Theorem guarantees this convergence. At fixed p strictly between 0 and 1, the standardized count approaches a standard normal as n grows. Rare-success cases may need much larger n.

Relationship to Bernoulli

A single Bernoulli trial is Binomial(1, p). The binomial is the sum of n independent Bernoullis — which is exactly why the CLT applies.

Compare the assumptions and event probabilities in Poisson vs binomial, or use the distribution chooser.

Use binomial vs hypergeometric to compare replacement rules directly, or follow the count-family relationship map to distinguish exact sums from Poisson limits.

Specify the counted event

For ten independent opportunities with common success probability 0.2, exactly two successes has probability C(10,2)×0.2²×0.8⁸≈0.301990. At most two includes zero and one as well, giving approximately 0.677800. The sequence view shows one run; the histogram repeats the entire ten-trial experiment.

The normal approximation matches mean np=2 and variance np(1−p)=1.6. To approximate P(X≤2), use the continuous cutoff 2.5. This continuity correction accounts for the width of the integer bar centered at 2. It does not repair severe skew or impossible negative values in the normal model; inspect the numerical error for the event you actually need.

Same margins, different dependence

Imagine drawing one Bernoulli(p) outcome and copying it into every trial. Each trial individually still succeeds with probability p, and the total still has mean np. But the count can only be zero or n, with variance n²p(1−p). Independent trials instead have variance np(1−p). Common probabilities are not enough: independence is a separate requirement.

If probabilities vary but trials stay independent, the sum is generally Poisson-binomial rather than binomial. Replacing the probabilities by their average preserves the mean but generally changes the distribution. Sampling without replacement introduces another dependence structure, as the hypergeometric comparison shows.

Reference

Random Services: binomial distribution derives the independent-indicator sum, PMF, and cumulative law. The examples here use its same-count convention.