Track 5 · Probability Distributions · Lesson 3 of 10
Geometric and hypergeometric distributions
Model the wait for a first success with the geometric distribution and sampling without replacement with the hypergeometric distribution, and know when the binomial is a safe shortcut.
A sales agent wants to know how many calls it usually takes to land the first sale; a quality inspector draws 10 panels from a lot of 50 and needs the chance of missing every defective one. Both questions look binomial at first glance, but neither is: one counts trials until a success, the other samples without replacement.
The binomial distribution fixes the number of trials in advance and counts the successes. Two close relatives change one of its ingredients:
- The geometric distribution keeps independent trials with a constant success probability , but stops at the first success and counts how many trials that took.
- The hypergeometric distribution keeps a fixed sample size , but draws without replacement from a finite population, so the trials are no longer independent.
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Calculate geometric probabilities, cdf values, means and SDs using the trials convention
- Explain the memoryless property and spot the gambler's fallacy
- Calculate hypergeometric probabilities for sampling without replacement
- Decide when the binomial is a safe approximation to the hypergeometric
- Choose between binomial, geometric and hypergeometric models from a description