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.
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
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.
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.
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.
Waiting time, batch sample, or event stream: the story picks geometric, hypergeometric, or Poisson.
Original lesson, flashcard and question revisions authored for this dataset.
Rewritten supplementary readings were written for this dataset. Includes original questions and videos by Khan Academy. Third-party license terms are unknown.
2 · Watch
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