---
title: "Discrete Models Beyond the Binomial"
description: "Geometric, hypergeometric and Poisson models, and what each assumes about the trials. Choosing a model is a modelling decision, not a formula lookup."
canonical: https://lightmysky.com/learn/mathematics/discrete-models-beyond-the-binomial-mt_9zc5xYjDg_
source: https://lightmysky.com/learn/mathematics/discrete-models-beyond-the-binomial-mt_9zc5xYjDg_.md
retrieved: 2026-09-12
---

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# Discrete Models Beyond the Binomial

Geometric, hypergeometric and Poisson models, and what each assumes about the trials. Choosing a model is a modelling decision, not a formula lookup.

Subject: Mathematics · Area: Probability · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/discrete-models-beyond-the-binomial-mt_9zc5xYjDg_

## Ready when they can

- Match a situation to the geometric, hypergeometric or Poisson model
- State the assumption that separates sampling with and without replacement
- Use the Poisson model as a limit of the binomial for rare events

## Lesson: Geometric, hypergeometric and Poisson

You choose a model by the story. Geometric: trials until the first success, each trial a fresh repeat, with mean 1 over p. Hypergeometric: successes in a fixed sample drawn without replacement from a finite batch of two types. Poisson: events in a fixed window when they are rare and roughly independent, the limit of a crowded binomial with rate lambda equals n times p.

**Example.** Roll a die until the first six: that wait is geometric. Draw 4 parts from a batch of 50 with 5 defectives and count defectives: no replacement from two types, so hypergeometric. Count calls arriving at a desk in an hour: rare, roughly independent arrivals in a window, so Poisson.

With replacement each draw faces the same chances, so trials stay independent. Without replacement each draw changes the mix, so chances shift and trials depend on each other. Cards from a deck without replacement are dependent: Tom is wrong to call each draw independent with the same chances throughout.

**Tip.** Match the question: waiting time, batch sample, or event stream. Set the Poisson rate to n times p: 1000 chips at 0.002 gives lambda 2. The mean geometric wait is 1 over p: p of 0.25 means 4 calls.

**Recap.** Waiting time, batch sample, or event stream: the story picks geometric, hypergeometric, or Poisson.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Conditional Probability, Independence and Bayes' Theorem](https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK)
- [The Binomial Distribution](https://lightmysky.com/learn/mathematics/the-binomial-distribution-mt_X2kVGK67Md)

## Opens up

- [Continuous Random Variables and Density Functions](https://lightmysky.com/learn/mathematics/continuous-random-variables-and-density-functions-mt_jlf4zDTeNm)
- [Stochastic Gene Expression: Bursting, Noise and Cell-to-Cell Variation](https://lightmysky.com/learn/science/stochastic-gene-expression-bursting-noise-and-cell-to-cell-variation-mt_YD6DOmrvng)