Direct comparison
Bernoulli vs Binomial Distribution Explained
Bernoulli vs binomial distribution: one trial versus a count over n trials, with side-by-side formulas, mean and variance, and a worked n = 10 example.
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How do Bernoulli distribution, Binomial distribution compare side by side?
The table below compares Bernoulli distribution, Binomial distribution across 11 procurement-relevant dimensions, from what it models through source.
Side-by-side comparison
| Dimension | Bernoulli distribution | Binomial distribution |
|---|---|---|
| What it models | The outcome of one trial with two possible results | The number of successes in n trials, each with two possible results |
| Parameters | p, the probability of a single success | n, the number of trials, and p, the probability of success on each trial |
| Possible values | k = 0 or k = 1 only | k = 0, 1, ..., n |
| Probability mass function | f(k) = 1 - p for k = 0; f(k) = p for k = 1 | f(k) = C(n,k) p^k (1 - p)^(n - k) for k = 0, 1, ..., n |
| Mean | p (computed as 0 x (1 - p) + 1 x p) | np |
| Variance | p(1 - p) (computed from E[X^2] - E[X]^2 = p - p^2) | np(1 - p), the square of the NIST standard deviation sqrt(np(1 - p)) |
| Relationship | The n = 1 case of the binomial: C(1,k) p^k (1 - p)^(1 - k) equals 1 - p at k = 0 and p at k = 1 | The distribution of the sum of n independent Bernoulli(p) variables (Bertsekas and Tsitsiklis, 2002) |
| Key assumption | A single trial with exactly two mutually exclusive outcomes | Exactly two mutually exclusive outcomes per trial, and a success probability p that is fixed for all trials (NIST) |
| Software parameters | SciPy scipy.stats.bernoulli takes p | SciPy scipy.stats.binom takes n and p; R binomial functions take size = n and prob = p |
| When to pick it | Each record is one yes/no outcome, for example one participant responding or not | You have a count of successes out of a known number of trials, for example 3 of 10 |
| Source | SciPy documentation, scipy.stats.bernoulli (read 2026-10-29) | NIST/SEMATECH e-Handbook, binomial distribution; SciPy scipy.stats.binom; R stats manual, Binomial (read 2026-10-29) |
