Binomial & Poisson Distribution Calculator
Choose binomial or Poisson; compute P(X=k), E(X) and Var(X), with cumulative probability.
Input Data
Results
At a glance:The binomial B(n,p) models the count of successes in n independent Bernoulli trials: P(X=k)=C(n,k)pᵏ(1−p)^{n−k}, E=np, Var=np(1−p). The Poisson Po(λ) models rare events per unit: P(X=k)=e^{−λ}λᵏ/k!, E=Var=λ; for large n, small p, binomial ≈ Poisson with λ≈np.
Formula
Binomial: P(X=k) = C(n,k) pᵏ (1−p)^{n−k}.
Binomial E(X)=np, Var(X)=np(1−p).
Poisson: P(X=k) = e^{−λ} λᵏ / k!.
Poisson E(X)=Var(X)=λ.
Approx: large n, small p ⇒ B(n,p) ≈ Po(np).
$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$$$P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}$$How to Use
- Choose the distribution (binomial / Poisson).
- Binomial: enter trials n and success p; Poisson: enter rate λ.
- Enter events k; the tool gives P(X=k), P(X≤k), E(X), Var(X).
- Compare with the case studies for hand-working.
Binomial vs Poisson
| Item | Binomial B(n,p) | Poisson Po(λ) |
|---|---|---|
| Use | n independent trials | rare events per unit |
| P(X=k) | C(n,k)pᵏ(1−p)^{n−k} | e^{−λ}λᵏ/k! |
| E(X) | np | λ |
| Var(X) | np(1−p) | λ |
Large n, small p ⇒ binomial ≈ Poisson (λ=np).
Case Studies
Binomial B(10, 0.3), k=3
P(X=3) = C(10,3)·0.3³·0.7⁷ ≈ 0.266828.
E=10·0.3=3, Var=10·0.3·0.7=2.1.
Coin B(5, 0.5), k=3 (heads)
P(X=3)=C(5,3)·0.5⁵=10/32=0.3125.
E=2.5, Var=1.25.
Poisson Po(3), k=2
P(X=2)=e^{−3}·3²/2! ≈ 0.224042.
E=Var=3.
Poisson Po(4), k=0 (zero events)
P(X=0)=e^{−4}≈0.018316 (e.g. machine failure).
P(X≤0)=0.018316.
Binomial ≈ Poisson
B(100, 0.01) vs Po(1): P(X=2)≈0.1844 vs 0.1839.
Check np=1=λ.
QC: defects Po(2)
Average 2 defects per item.
P(X≥3)=1−P(X≤2)=1−0.6767=0.3233.
FAQ
When to use binomial vs Poisson?
Binomial for a fixed number n of independent trials (e.g. count of defectives in a sample); Poisson for rare events per unit time/space (calls, failures).
How to approximate?
When n is large and p small (np moderate, e.g. n≥20, p≤0.05), B(n,p) can be approximated by Po(λ=np) for easier computation.
What are E(X) and Var(X)?
E(X) is the mean (long-run expected count); Var(X) measures spread. For Poisson both equal λ; for binomial Var=np(1−p)≤E.
P(X≤k) vs P(X=k)?
P(X=k) is exactly k; P(X≤k) is cumulative (sum 0..k), used for 'no more than k' probabilities.
Can p exceed 1?
No. Binomial success p is between 0 and 1 inclusive; Poisson λ is any non-negative real.
Must k be integer?
Yes, both binomial and Poisson count non-negative integer occurrences.
Why Var=E for Poisson?
Equality of mean and variance is a mathematical property; if real data shows Var≫E (over-dispersion), Poisson may be a poor fit.
What can it model?
Call-center arrivals, web traffic, radioactive counts, accident numbers — rare, independent events are commonly Poisson-modeled.
Related Tools
References
Content review: Calculatorism Science Team. Binomial/Poisson PMF, E(X), Var(X) and cumulative probability verified. Results are for reference only.