--- title: "The Addition Rule and Mutually Exclusive Events" description: "Use P(A or B) = P(A) + P(B) - P(A and B), and see why the subtraction disappears when two events cannot happen together." canonical: https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE source: https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Addition Rule and Mutually Exclusive Events Use P(A or B) = P(A) + P(B) - P(A and B), and see why the subtraction disappears when two events cannot happen together. Subject: Mathematics · Area: Probability · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE ## Ready when they can - Decide whether two events are mutually exclusive before adding - Apply the addition rule and subtract the overlap where there is one - Use the fact that all outcomes sum to 1 to find a missing probability - Distinguish independent events from mutually exclusive events ## Lesson: Two questions at once, without double counting Counting all outcomes gave you the bottom number of a probability. When an event has two descriptions, counting the top number needs more care. The prize table has 40 items, all equally likely to be handed out. Sixteen are toys, ten are red, and four are red toys included in both counts. Somebody asks for the chance of getting a toy or something red. In probability, 'or' means at least one of the two, including both. Adding the counts is an obvious first step. *(drawing: The third bar is not a separate pile. Those four prizes are already standing inside the first bar and inside the second.)* Adding gives 16 + 10 = 26, but this counts the four red toys twice: once as toys and once as red prizes. Subtract one copy: 16 + 10 - 4 = 22 prizes are a toy or red or both. As a probability, 22 / 40 = 0.55. The rule is P(A or B) = P(A) + P(B) - P(A and B). The difference between the plain sum and the correct probability is the overlap. So P(A and B) = P(A) + P(B) - P(A or B). *(drawing: Twenty-two prizes, not twenty-six. Whatever sits in both piles has to be given back once.)* Sometimes there is nothing to subtract. Each of the 40 prizes has exactly one size: 18 small, 14 medium and 8 large. No prize is both small and large, so those events are mutually exclusive with no overlap. The rule's last term is zero, so add directly: P(small or large) = 0.45 + 0.2 = 0.65. **Example.** Every prize has one size and no other size exists, so the sizes cover the table with no overlaps. Their probabilities total 1: 0.45 + 0.35 + 0.2 = 1. Use this total to find a missing probability: large is 1 - 0.45 - 0.35 = 0.2. The same idea gives the chance of an event not happening: 1 minus the chance that it does. Check overlap: can one outcome belong to both events? Independence asks whether learning one happened changes the other's chance. Its test is P(A and B) = P(A) × P(B). Mutually exclusive events with positive probabilities are dependent; if either has probability 0, they are independent. Cards 1 to 12: above 6 and below 10 cover all cards but share 7, 8 and 9. Missing a real overlap makes an 'or' answer too big; subtracting an absent overlap makes it too small. **Recap.** P(A or B) = P(A) + P(B) - P(A and B). Exclusive events share nothing, so the last term is zero. Outcomes covering everything without overlap add to 1. Independent is a different idea. ## Practice 34 questions on this page, each with its working shown. ## Needs first - [Probabilities Sum to One](https://lightmysky.com/learn/mathematics/probabilities-sum-to-one-mt_3fwYu7imd4) - [Counting Outcomes with the Product Rule](https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p) ## Opens up - [Probability Spaces: Sample Spaces, Events and Axioms](https://lightmysky.com/learn/mathematics/probability-spaces-sample-spaces-events-and-axioms-mt_4gsBeqJtIQ) - [Independent Events and the Multiplication Rule](https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN)