---
title: "Addition Rule — AP Stats Definition & Formula Guide"
description: "The addition rule gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B), the probability of A or B. Learn how it's tested in AP Stats Unit 4 and why you subtract the overlap."
canonical: "https://fiveable.me/ap-stats/key-terms/addition-rule"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 2"
---

# Addition Rule — AP Stats Definition & Formula Guide

## Definition

In AP Statistics, the addition rule says the probability that event A or event B (or both) occurs is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). You subtract P(A ∩ B) because adding P(A) and P(B) counts the overlap twice (Topic 4.6, VAR-4.E.4).

## What It Is

The addition rule is the formula for the [probability](/ap-stats/unit-2/intro-probability/study-guide/gfnBWfyMANOxF3vWLrbA "fv-autolink") of a **union** of two events, the [chance](/ap-stats/unit-3 "fv-autolink") that A happens, or B happens, or both happen. It states that P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Here's the intuition. Picture a Venn diagram. If you just add P(A) and P(B), the overlapping middle region gets counted twice, once inside circle A and once inside circle B. Subtracting P(A ∩ B) removes that double count. That's the entire logic of the rule. Two special cases follow immediately. If A and B are **[mutually exclusive](/ap-stats/unit-2/mutually-exclusive-events/study-guide/iBljImMDLJ8bSWeWXbi6 "fv-autolink")** (they can't both happen), the overlap is zero and the rule shrinks to P(A ∪ B) = P(A) + P(B). If A and B are **independent**, you can compute the overlap as P(A) ⋅ P(B) first (per VAR-4.E.2), then plug it into the addition rule.

## Why It Matters

The addition rule lives in **[Unit 4](/ap-stats/unit-4 "fv-autolink"): Probability, Random Variables, and Probability Distributions**, specifically **Topic 4.6 (Independent Events and Unions of Events)**. It directly supports learning objective **4.6.A**: calculate probabilities for independent events and for the union of two events, with the rule itself spelled out in essential knowledge **VAR-4.E.4**. Unit 4 is the bridge between the data analysis of Units 1-3 and the inference of Units 6-9, and the addition rule is one of the handful of probability formulas you'll use constantly. It also appears on the [AP Stats](/ap-stats "fv-autolink") formula sheet, so your job isn't memorization. Your job is knowing when a question is asking for an "or" probability and remembering not to double-count the overlap.

## Connections

### [Probability of union (Unit 4)](/ap-stats/key-terms/probability-of-union)

The addition rule and the probability of a union are two sides of the same coin. P(A ∪ B) is the quantity you want, and the addition rule is the recipe for computing it. Anytime a problem says "or," you're in union territory.

### Independence and the multiplication rule (Unit 4)

[Independence](/ap-stats/key-terms/independence "fv-autolink") (Topic 4.6) is how you find the P(A ∩ B) piece when it isn't handed to you. If A and B are independent, P(A ∩ B) = P(A) ⋅ P(B), and you feed that result straight into the addition rule. Many exam questions chain these two rules together in exactly that order.

### [Law of total probability (Unit 4)](/ap-stats/key-terms/law-of-total-probability)

Both rules are about building one probability from pieces. The addition rule combines two overlapping events, while the [law of total probability](/ap-stats/key-terms/law-of-total-probability "fv-autolink") breaks an event into non-overlapping cases and adds them up. Together they're your toolkit for assembling probabilities from parts.

### Two-way tables and conditional probability (Unit 4)

Problems like "35% did community service, 42% played sports, 15% did both" can be solved with the formula or by filling in a [two-way table](/ap-stats/key-terms/two-way-table "fv-autolink") or Venn diagram. The table makes the double-counted overlap visible, which is why it's a great check on your addition-rule arithmetic.

## On the AP Exam

The addition rule is a multiple-choice staple. Stems typically come in two flavors. The first hands you all three pieces directly, like a study where 40% of students own a car, 25% own a bicycle, and 15% own both, and asks for the probability a student owns either or both (answer: 0.40 + 0.25 − 0.15 = 0.50). The second gives you independent events, like P(A) = 0.3 and P(B) = 0.4 with A and B independent, and expects you to compute the overlap first: P(A ∩ B) = (0.3)(0.4) = 0.12, so P(A ∪ B) = 0.3 + 0.4 − 0.12 = 0.58. No released FRQ has hinged on the term "addition rule" by name, but probability FRQs routinely require an "or" calculation as one step, and showing the formula with your numbers plugged in is how you earn that component of the score. The formula is on the provided formula sheet, so the real test is recognizing "or" language and remembering to subtract the intersection.

## addition rule vs Multiplication rule

The addition rule is for "or" (unions); the multiplication rule is for "and" (intersections). P(A ∪ B) = P(A) + P(B) − P(A ∩ B) tells you the chance at least one event happens, while P(A ∩ B) = P(A) ⋅ P(B | A) tells you the chance both happen. They often appear in the same problem, since you may need the multiplication rule to find the overlap before the addition rule can finish the job. Quick check: if the question says "or" or "at least one," add and subtract the overlap; if it says "and" or "both," multiply.

## Key Takeaways

- The addition rule states P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and it answers any question asking for the probability of A or B or both.
- You subtract P(A ∩ B) because adding P(A) and P(B) counts the overlap region of the Venn diagram twice.
- If events are mutually exclusive, the overlap is zero and the rule simplifies to P(A ∪ B) = P(A) + P(B).
- If events are independent, first compute the overlap as P(A ∩ B) = P(A) ⋅ P(B), then plug it into the addition rule.
- Mutually exclusive and independent are not the same thing, and in fact two events with nonzero probabilities cannot be both.
- The formula is on the AP Stats formula sheet, so the exam tests whether you recognize "or" situations and avoid double-counting, not whether you memorized it.

## FAQs

### What is the addition rule in AP Stats?

It's the formula for the probability of a union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). It gives the chance that event A or event B or both occur, and it's covered in Topic 4.6 of Unit 4 (essential knowledge VAR-4.E.4).

### Why do you subtract P(A and B) in the addition rule?

Because outcomes where both A and B happen get counted twice, once in P(A) and once in P(B). Subtracting P(A ∩ B) removes the duplicate count. In a Venn diagram, it's the overlapping middle region.

### Can I just add the probabilities without subtracting anything?

Only if the events are mutually exclusive, meaning they can't both happen, so P(A ∩ B) = 0. For example, with 40% car owners, 25% bike owners, and 15% owning both, the answer is 0.40 + 0.25 − 0.15 = 0.50, not 0.65. Forgetting the subtraction is the single most common error on these questions.

### What's the difference between the addition rule and the multiplication rule?

Addition is for "or" and multiplication is for "and." The addition rule finds P(A ∪ B), the chance at least one event occurs, while the multiplication rule finds P(A ∩ B), the chance both occur. Problems often use both, like computing P(A ∩ B) = (0.4)(0.3) = 0.12 for independent events, then P(A ∪ B) = 0.4 + 0.3 − 0.12 = 0.58.

### Is the addition rule on the AP Stats formula sheet?

Yes, P(A ∪ B) = P(A) + P(B) − P(A ∩ B) appears on the formula sheet you get with the exam. You don't need to memorize it, but you do need to recognize when a question is asking for an "or" probability and apply it correctly.

## Related Study Guides

- [2.7 Independent Events and Unions of Events](/ap-stats/unit-2/independent-events-unions-events/study-guide/aMOuOhtDQIJAtgZx3JNz)

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