Addition Rule Formula

The addition rule finds the probability that at least one of two events occurs.

The Formula

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

When to use: If you want “A or B,” start by adding A and B. Then fix the double-counting by removing the part that belongs to both events.

Quick Example

From a standard deck, the probability of drawing a heart or a face card is 13/52+12/52−3/52=22/5213/52 + 12/52 - 3/52 = 22/52 because the Jack, Queen, and King of hearts were counted twice.

Notation

A∪BA \cup B means “A or B,” including the case where both happen.

What This Formula Means

The addition rule finds the probability that at least one of two events occurs. It adds the probabilities of the two events and then subtracts any overlap so the shared outcomes are not counted twice.

If you want “A or B,” start by adding A and B. Then fix the double-counting by removing the part that belongs to both events.

Formal View

The addition rule corrects for inclusion-exclusion on two sets by subtracting the intersection once after adding the marginal probabilities.

Worked Examples

Example 1

easy
P(A)=0.5P(A)=0.5, P(B)=0.5P(B)=0.5, P(A∩B)=0.5P(A\cap B)=0.5. Find P(A∪B)P(A\cup B) and interpret.

Answer

0.50.5

First step

1
P(A∪B)=0.5+0.5−0.5=0.5P(A\cup B)=0.5+0.5-0.5=0.5.

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Example 2

medium
Two dice are rolled. Find P(sum is 7 or sum is 11)P(\text{sum is 7 or sum is 11}).

Example 3

hard
Among 120 students, 60 take Math, 45 take Physics, 50 take Chemistry; 20 take both Math and Physics, 25 Math and Chemistry, 15 Physics and Chemistry, and 10 take all three. How many take at least one subject?

Common Mistakes

  • Adding probabilities without subtracting overlap - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Using the addition rule for “and” problems - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Assuming the overlap is zero without checking the context - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Choosing addition rule from a keyword alone - Keywords like chance, probability, outcome are only clues; the data structure must match the concept.

Why This Formula Matters

Addition Rule helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

Frequently Asked Questions

What is the Addition Rule formula?

The addition rule finds the probability that at least one of two events occurs. It adds the probabilities of the two events and then subtracts any overlap so the shared outcomes are not counted twice.

How do you use the Addition Rule formula?

If you want “A or B,” start by adding A and B. Then fix the double-counting by removing the part that belongs to both events.

What do the symbols mean in the Addition Rule formula?

A∪BA \cup B means “A or B,” including the case where both happen.

Why is the Addition Rule formula important in Statistics?

Addition Rule helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

What do students get wrong about Addition Rule?

Students often know a procedure related to addition rule but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

What should I learn before the Addition Rule formula?

Before studying the Addition Rule formula, you should understand: compound events, stat sample space.

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