Addition Principle

Definition

If a task can be performed in one of several mutually exclusive ways, where the first way has $n$ outcomes and the second way has $m$ outcomes, the total number of ways to perform the task is the sum of the outcomes: $N = n + m$ How to read: “The total N equals n plus m.” Meaning / when to use: Used when choices are mutually exclusive alternatives (either path A OR path B, but not both).

Why It Matters

It is essential for partitioning complex counting problems into simpler, disjoint cases, ensuring that no possibilities are missed and none are counted multiple times.

Core Concepts

• Mutually Exclusive (Disjoint): The options cannot happen simultaneously. Choosing one option rules out all others.
• Sum Rule: The total size of the union of pairwise disjoint sets is the sum of their individual sizes.
• OR Operator: In logic and probability, the word “or” corresponds to addition when events are disjoint.

Connected Concepts

• Multiplication Principle
• Inclusion-Exclusion Principle
• Permutations
• Combinations
• Probability Models: Equally Likely Outcomes
• First Principles Thinking