Interactive module guide

Binomial to Poisson Approximation Guide

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The binomial-to-Poisson approximation covers rare events: when the number of trials n is large and the per-trial success probability p is small, a Binomial(n, p) count behaves almost exactly like a Poisson count with the same mean λ = np.

This page holds λ fixed on one slider while n grows (and p = λ/n shrinks) on the other, so you can watch the binomial bars settle onto the Poisson curve.

The limit is the law of rare events, and it explains why defect counts, accident tallies, and other many-opportunities-but-low-chance phenomena are modeled directly as Poisson without ever knowing n and p separately.

Is this the right module for my question?

Use it when

Use this view to justify replacing a Binomial(n, p) model by Poisson(np) when n is large and p is small—typically p at or below 0.05 with np moderate—so that only the event rate needs to be estimated or communicated.
Use it to teach why the Poisson distribution appears in nature: many independent opportunities, each with a tiny probability, produce Poisson-distributed totals regardless of the exact number of opportunities.
Use it to sanity-check software or textbook shortcuts: sliding n from 20 to 500 at fixed λ shows concretely how fast the pmf difference decays, and which counts (near zero and near the mean) converge first.

Choose another method when

Avoid the approximation when p is not small. With p near one half, the binomial is nearly symmetric while a Poisson with the same mean is right-skewed with variance equal to its mean; the normal approximation is the appropriate one there.
Avoid it when n is genuinely small: with only a handful of trials the exact binomial is trivial to compute, and no approximation earns its error.
Avoid it when the trials are dependent or the success probability varies across trials; both binomial and Poisson reasoning fail together, and an overdispersed model such as the negative binomial is usually the honest replacement.

Interactive tool

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Example preview

Rare defects approaching Poisson

Inputs: Hold λ=np at 3 while n grows from 20 to 500 and p shrinks.
Representative result: At n=100 (p=0.03), P(X=0) is 0.0476 exactly versus 0.0498 under Poisson(3); the gap keeps shrinking with n.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

2:27 min

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Read the complete transcript

Why the Binomial Becomes Poisson. The binomial becomes Poisson along one specific rare-event path: the number of trials grows, the success probability shrinks, and their product lambda stays fixed. The trials still need to be independent, with one common probability p. Many opportunities and a tiny chance per opportunity create the Poisson count. Hold lambda equal to n times p. If n rises, p must fall as lambda divided by n. Holding p fixed would be a different limit. Both models then have mean lambda. But the binomial variance is lambda times one minus p, just below the Poisson variance lambda until p nearly vanishes. At every displayed count, the page evaluates both probability mass functions exactly. No simulated sample is needed to watch the bars settle together. Zero events gives a sensitive checkpoint: binomial uses one minus p to the n, while Poisson uses e to the minus lambda. Their ratio approaches one. A small p is a practical starting rule, not a guarantee for every tail. The page rates the regime and keeps exact binomial probabilities available.

For a concrete batch, set n to one hundred and p to point zero three. Both distributions have mean lambda equal to three, while their variances are two point nine one and three. The exact probability of zero defects is point zero four seven six. Poisson three gives point zero four nine eight, only about two thousandths higher. For at most one defect, the binomial probability is point one nine four six, versus point one nine nine one under Poisson. For six or more defects, the exact upper tail is point zero eight zero eight, versus point zero eight three nine. The decision-relevant tail still deserves inspection. Across the full support, total variation is about point zero zero seven six. The educational n p squared bound here is point zero nine, safely above the observed error. Do not use this shortcut for moderate p, dependent trials, clustered events, or a drifting rate. Those failures can break both the binomial and simple Poisson stories. Keep lambda fixed, watch p shrink, and confirm the probability that matters. Explore the binomial-to-Poisson limit free at Distri Scope dot com.

How to read the result

Watch the variance while sliding: the binomial variance np(1−p) climbs toward the Poisson variance λ as p shrinks, and the small remaining gap is the visible part of the approximation error.

The zero-count probability is a sensitive indicator: the binomial gives (1−p)ⁿ against the Poisson e^{−λ}, and their ratio approaches one precisely as p → 0.

If the two shapes disagree noticeably at your parameter values, the rating panel will usually already say fair or poor; trust the exact binomial in that regime.

Inputs and parameter meaning

The main slider sets the number of trials n from 20 to 500; the rate slider sets λ = np from 1 to 10, and the per-trial probability p = λ/n is displayed in the chart legend.

Because λ is held fixed, moving n changes p in the opposite direction—that coupled motion is the defining structure of the rare-event limit.

The chart truncates the display near λ + 4√λ, where both pmfs are already negligible; the full distributions extend further with vanishing mass.

Limit, parameter, and conditions

X_n \sim \operatorname{Bin}(n, \lambda/n) \xrightarrow{d} \operatorname{Poisson}(\lambda) \quad \text{as } n \to \infty

Interactive parameter: Trial count n with the mean λ = np held fixed.

The trials are independent with a common small success probability p.
The comparison holds n·p fixed at λ while n grows and p shrinks.
The approximation is strongest when p is small (roughly p ≤ 0.05) and n is large.

What the method is doing

The binomial pmf is evaluated exactly at every displayed count and compared with the Poisson pmf at mean λ; nothing is simulated.

The displayed overlap measures the shared area of the two pmfs across the plotted counts, an educational summary rather than a formal error bound; the classical Le Cam bound guarantees total-variation error at most np² for the full distributions.

The rule-of-thumb rating follows the common teaching guidance: good when n ≥ 100 with p ≤ 0.01, fair when n ≥ 20 with p ≤ 0.05, poor otherwise.

Worked example: defects at λ = 3 as n grows

A production process yields defective units at an average rate of λ = 3 per batch. Compare the exact Binomial(n, 3/n) distribution with the Poisson(3) approximation for a batch of n = 100 units (so p = 0.03), focusing on the zero-defect probability, at most one defect, and the upper tail of six or more.

1Zero defects: the exact binomial gives 0.97¹⁰⁰ ≈ 0.0476 while Poisson(3) gives e⁻³ ≈ 0.0498—a difference of about 0.002.
2At most one defect: the exact probability is ≈ 0.1946 against the Poisson ≈ 0.1991.
3Six or more defects: the exact upper tail is ≈ 0.0808 against the Poisson ≈ 0.0839; tail errors stay in the third decimal place.
4Summed over all counts, half the total absolute pmf difference is ≈ 0.0076, comfortably below the Le Cam bound np² = 100 × 0.03² = 0.09.

Interpretation

At n = 100 and p = 0.03 every practical probability agrees to about two decimal places, so modeling the batch as Poisson(3) is safe—and it frees the analysis from needing n and p separately. Sliding n down to 20 (p = 0.15) visibly bends the bars away from the curve, which is exactly the regime the rating panel labels poor.

Common mistakes

Do not fix p and grow n—that route leads to the normal approximation, not the Poisson one. The rare-event limit requires p to shrink as n grows with np stable.
Do not read the Poisson replacement as exact for moderate p even when the chart looks close; probabilities in the far tails converge more slowly than the visible bars.
Do not use either model when events cluster (dependence) or the rate drifts across trials; agreement between binomial and Poisson does not validate the independence assumption they share.

Limits and independent validation

The overlap statistic covers only the plotted counts; formal error control comes from bounds such as Le Cam’s np², which the page reports conceptually but does not compute per parameter.

The sliders cap n at 500 and λ at 10—enough to see the convergence pattern, though real applications may involve far larger n and smaller p.

The page compares distributions, not data; whether a Poisson model fits observed counts is a question for the data-fitting tool.

Before using the result

Recompute one probability by hand: the zero-count values (1−p)ⁿ and e^{−λ} are one-line calculations that anchor the whole comparison.
Check the mean and variance panel: both distributions share mean λ, and the binomial variance λ(1−p) should sit just below the Poisson variance λ.
Confirm the rating boundaries by sliding across them: the visual gap between bars and curve should change character as p crosses 0.05.

binomial to normal →See the complementary limit where p stays fixed and the binomial becomes normal.Poisson distribution →Work with the limiting distribution directly, including exact tail probabilities.negative binomial distribution →Model rare-event counts whose variance exceeds the Poisson prediction.

← Chi-square → Normal (Large df)Poisson Process (Counts, Gaps, Arrivals) →

Explore Other Convergence Visualizations

See every option in the Distribution Convergence Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.