--- title: "Discrete Models Beyond the Binomial" description: "Geometric, hypergeometric and Poisson models make different assumptions. Choosing a model means deciding which assumptions fit the story before using any formula." 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Discrete Models Beyond the Binomial Geometric, hypergeometric and Poisson models make different assumptions. Choosing a model means deciding which assumptions fit the story before using any formula. 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 Geometric counts independent trials until the first success, with fixed chance p. First success on trial k has chance (1 - p) to the power k - 1, times p; the mean is 1 / p. Hypergeometric counts successes in a random sample without replacement from a finite two-type batch. Poisson counts events in a fixed window. For arrivals, assume a constant rate, independent counts in disjoint intervals and negligible chance of multiple arrivals in a very short interval. *(drawing: A geometric wait of 4 trials: three misses, then a hit. With success chance p that has chance (1 - p) × (1 - p) × (1 - p) × p.)* **Example.** Assume independent free throws with p = 0.4. The chance of first scoring on shot 3 is 0.6 × 0.6 × 0.4 = 0.144; the mean wait is 1 / 0.4 = 2.5 shots. Randomly pick 2 pens without replacement from 12 pens, of which 3 are dried out: the dried-out count is hypergeometric. Poisson may model typing errors on a page if disjoint regions have independent counts, the rate per unit area is constant and multiple errors in a tiny region are negligible. *(drawing: 12 pens, 3 dried out. After one pick the mix changes, so the next pick has different chances.)* With replacement each draw faces the same chances, so trials stay independent. Without replacement each draw changes what is left. Take 2 pens from the box above. If the first is dried out, 11 pens remain and 2 are dried out, so the second draw has chance 2 / 11, not 3 / 12. The draws depend on each other, so the binomial and geometric models do not fit. For n independent trials with fixed success chance p, the count is binomial. When n is large and p is small, use a Poisson approximation with lambda = n × p. A tray of 600 seeds, failing independently with chance 0.005 each, gives lambda = 3. In the binomial limit, n grows and p shrinks while n × p stays fixed. For a constant-rate arrival process, lambda is the average arrival rate times the window length. **Recap.** Waiting time, batch sample, or event stream: the story picks geometric, hypergeometric, or Poisson. *Original lesson, flashcard and question revisions authored for this dataset.* ## 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)