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.

Open the interactive lesson All topics

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.

See the full formula reference

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.

Browse the full glossary

Go interactive

Work this topic in StatRise

This module opens on its first lesson free; the rest of it unlocks with Premium. Every calculator and simulation below is free.

Probability Foundations lessons (first free)Probability calculators Combinatorics calculators Coin & Dice simulation Practice questions

Keep reading