Direct comparison
Binomial vs Geometric Distribution: Which Fits?
Binomial vs geometric: fixed trials versus trials to first success. Side-by-side pmf, mean, variance, SciPy vs R conventions and a worked example at p = 0.3.
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How do Binomial distribution, Geometric distribution compare side by side?
The table below compares Binomial distribution, Geometric distribution across 12 procurement-relevant dimensions, from what the random variable counts through source.
Side-by-side comparison
| Dimension | Binomial distribution | Geometric distribution |
|---|---|---|
| What the random variable counts | The number of successes in a fixed number n of independent trials. | The number of independent trials until the first success (OpenStax convention: X = 1, 2, 3, ...). R instead counts the failures before the first success. |
| What is fixed in advance | The number of trials, n. You stop after n trials whatever happened. | The stopping rule: you keep going until the first success. The number of trials is the outcome, not a design choice. |
| Possible values | k = 0, 1, ..., n (SciPy binom). The count can never exceed n. | k = 1, 2, 3, ... with no upper limit (SciPy geom, OpenStax). R dgeom starts at x = 0 because it counts failures. |
| Probability mass function | C(n, k) p^k (1 − p)^(n − k) | Trials convention (SciPy): (1 − p)^(k − 1) p. Failures convention (R): p (1 − p)^x. |
| Parameters | Two: n (number of trials) and p (probability of success on one trial). | One: p (probability of success on one trial). |
| Mean | np | 1/p when counting trials (OpenStax). Counting failures instead, the mean is one lower: (1 − p)/p. |
| Variance | np(1 − p) | (1 − p)/p², the same under either counting convention because shifting by one does not change spread. |
| Assumptions shared by both | Each trial has two outcomes, the same success probability p, and independent trials. | The same: Bernoulli trials with a constant p for every trial. |
| Worked example at p = 0.3 | P(exactly 1 success in 3 trials) = 3 × 0.3 × 0.7² = 0.441. | P(first success on trial 3) = 0.7² × 0.3 = 0.147. Same p, different question, different answer. |
| Software parameters | SciPy binom.pmf(k, n, p); R dbinom(x, size, prob). | SciPy geom.pmf(k, p) with k ≥ 1 (trials); R dgeom(x, prob) with x ≥ 0 (failures). To match SciPy k, pass x = k − 1 to R. |
| When to pick each | Pick the binomial when the number of trials is set in advance and you count successes, such as the number of positive samples among 20 tested. | Pick the geometric when you repeat until the first success and the quantity of interest is how long that takes, such as the trial on which the first positive appears. |
| Source | SciPy scipy.stats.binom; R stats Binomial; NIST/SEMATECH e-Handbook of Statistical Methods; OpenStax Introductory Statistics (read 2026-10-29). | SciPy scipy.stats.geom; R stats Geometric; OpenStax Introductory Statistics section 4.4 (read 2026-10-29). |
Common questions
Common questions about Binomial distribution vs Geometric distribution
What is the main difference between binomial and geometric distributions?
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The binomial fixes the number of trials and counts successes. The geometric fixes the stopping rule (stop at the first success) and counts how many trials it takes. Both assume independent trials with the same success probability p.
Is the geometric distribution a special case of the negative binomial?
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Yes. The negative binomial counts trials or failures until r successes, and r = 1 gives the geometric distribution. See the CASRAI guide on the negative binomial distribution.
Why does the geometric formula sometimes use (1 − p)^(k − 1) and sometimes (1 − p)^k?
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The exponent depends on what is counted. SciPy and OpenStax count trials (k = 1, 2, ...), giving (1 − p)^(k − 1) p. R counts failures before the first success (x = 0, 1, ...), giving p(1 − p)^x. They describe the same distribution shifted by one.
Can the same data be described by both distributions?
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Not for the same variable. One run of trials until the first success gives one geometric observation. If you fix n trials and count successes, that count is binomial. The two answer different questions even at the same p.
