Track 4 · Probability · Lesson 7 of 9
Counting: permutations and combinations
Count outcomes with the multiplication principle, permutations and combinations, and use the counts to find probabilities.
A quality inspector takes 5 items from a batch of 30 that secretly contains 3 faulty ones. What is the chance the sample contains no faulty item at all, so the batch passes? When outcomes are equally likely, a probability is a ratio of two counts — and getting the counts right is the whole problem.
When every outcome is equally likely,
For two dice you can list all 36 outcomes. For a sample of 5 items from 30 you cannot: there are over a hundred thousand possible samples. This lesson gives you three counting tools and, more importantly, a way to decide which one fits.
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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Use the multiplication principle for choices made in stages
- Decide whether order matters and whether repeats are allowed before choosing a counting rule
- Compute permutations and combinations and explain why they differ by a factor of k!
- Find probabilities by counting, including acceptance-sampling and committee problems
- Explain the birthday problem by counting pairs