When should I use Poisson vs Binomial?

Use Poisson for events occurring at a rate over time/space. Use Binomial for fixed trials with known probability.

Why does mean equal variance in Poisson?

This is a fundamental property: E[X] = Var(X) = λ. If variance ≠ mean in your data, consider negative binomial instead.

As n→∞ and p→0 with np=λ constant, Binomial(n,p) → Poisson(λ). Use Poisson to approximate Binomial when n>20 and p<0.05.

The number of trials needed to get the first success. Examples: coin flips until heads, interviews until job offer.

P(X > m+n | X > m) = P(X > n). Past failures don't affect future probability-each trial is independent.

Geometric is NegBin(1,p). Negative Binomial counts trials until r successes; Geometric is the special case r=1.

Calculate discrete probability distributions: Binomial, Poisson, and Geometric with step-by-step solutions.

n (number of independent trials)

k (number of successes)

p (probability of success per trial)

Formula

P(X = 3) = 10!3! × 7! × 0.5^3 × (1-0.5)^7

Calculate probability for:

Enter parameters and click Calculate to see the probability.

Fixed number of trials (n), only two outcomes (success/failure), constant probability (p), and independent trials.

When n×p ≥ 5 and n×(1-p) ≥ 5, use Normal(μ=np, σ²=np(1-p)) for approximation.

Bernoulli is a single trial (n=1). Binomial is the sum of n independent Bernoulli trials.