Discrete Models Beyond the Binomial · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Geometric, hypergeometric and Poisson

Mathematics · Probability · ages 18-19
Name ______________________   Date ____________

  1. A batch of 50 parts holds 5 defectives. Four parts are drawn without replacement and defectives are counted. Which model fits?

    • Hypergeometric
    • Geometric
    • Poisson
    • Exponential

  2. A fair die is rolled until the first six appears, and the number of rolls is recorded. Which model fits?

    • Geometric
    • Hypergeometric
    • Poisson
    • Binomial

  3. A fair die is rolled until the first six appears, and the number of rolls is recorded. Which model fits?

    • Poisson
    • Geometric
    • Hypergeometric

  4. Cards are drawn from a deck without replacement. Tom says each draw is independent with the same chances throughout. Is Tom right?

    Circle one:   True   False

  5. Each sales call succeeds with chance 0.25, independently. How many calls are expected before the first success?

    Answer: ______________

  6. Each sales call succeeds with chance 0.25, independently. How many calls are expected before the first success?

    Answer: ______________

  7. A coin with P(heads) = 0.5 is tossed until the first head. What is the chance the first head lands on toss 3?

    Answer: ______________

  8. A coin with P(heads) = 0.5 is tossed until the first head. What is the chance the first head lands on toss 3?

    Answer: ______________

  9. Which model fits the number of calls arriving at a desk in an hour, and why?

    • Geometric, since the hour waits for a first call
    • Poisson, since arrivals are rare and roughly independent in a window
    • Hypergeometric, since hours are drawn without replacement

  10. Kim draws parts from one batch without replacement and treats each draw as facing the same chances. Is Kim right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Geometric, hypergeometric and Poisson W1-mt_9zc5xYjDg_-s1

  1. Hypergeometric · Sampling without replacement from a finite batch with two types is the hypergeometric story.
  2. Geometric · Counting trials until the first success, with repeats under identical conditions, is the geometric story.
  3. Geometric · Counting trials until the first success, with fresh repeats, is the geometric story.
  4. False · Tom is wrong: drawn cards leave the deck, so later chances shift and trials depend.
  5. 4 · The mean geometric wait is 1 divided by p: 1 divided by 0.25 is 4 calls.
  6. 4 · The geometric mean waiting time is 1 / p = 1 / 0.25 = 4 calls.
  7. 0.125 · Two failures then a success: 0.5 x 0.5 x 0.5 = 0.125.
  8. 0.125 · Two failures then a success: 0.5 times 0.5 times 0.5 is 0.125.
  9. Poisson, since arrivals are rare and roughly independent in a window · An event stream of rare, roughly independent arrivals in a fixed window is the Poisson story.
  10. False · Kim is wrong: each draw changes the mix, so chances shift and draws depend on each other.
Worksheet · LightMySky