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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Waiting time, batch sample, or event stream: the story picks geometric, hypergeometric, or Poisson.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.