Module 2 · Probability Foundations · Topic 9

Counting Principles

Fundamental Counting Principle. If one event can occur in m ways and a second independent event can occur in n ways, then the two events together can occur in m × n ways.

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Key formulas

Factorial

n! = n × (n − 1) × (n − 2) × … × 1, with 0! = 1

From the Probability Foundations formula sheet

Conditional Probability: P(A | B) = P(A and B) / P(B) — Probability of A after restricting attention to cases where B occurred.
Bayes' Theorem: P(A | B) = P(B | A)P(A) / P(B) — Updates a prior probability after observing evidence.

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Worked example

License Plate Example

A license plate has 3 letters followed by 4 digits. Total possibilities = 26³ × 10⁴ = 17,576 × 10,000 = 175,760,000.

When to use it

Tree Diagrams Help

For small problems, drawing a tree diagram makes counting systematic. Each branch represents one stage of the process.

Related glossary terms

Independence: Two events are independent when one occurring does not change the probability of the other. Example: Two separate fair coin flips are independent.

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