--- 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: 2026-09-12 --- > **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 outcomes gave you the bottom of a probability. Now the top gets harder, because people ask two things at once. The prize table has 40 items on it, all equally likely to be handed out. Sixteen of them are toys. Ten of them are red. Four are red toys, so they are in both counts. Somebody asks the chance of getting a toy or something red. In probability that 'or' means at least one of the two, which takes in both, and the obvious move is to add. *(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, and that is wrong. The four red toys were counted once as toys and once as red, so 26 counts them twice. Take one copy back: 16 + 10 - 4 = 22 prizes are a toy or red or both. As probabilities, 22 / 40 = 0.55. The rule behind it is P(A or B) = P(A) + P(B) - P(A and B), and the gap between the plain sum and the truth is the overlap itself. *(drawing: Twenty-two prizes, not twenty-six. Whatever sits in both piles has to be given back once.)* Sometimes there is nothing to give back. Each of the 40 prizes has exactly one size: 18 small, 14 medium and 8 large. Small and large cannot share a prize, so they are mutually exclusive with no overlap. The rule loses its last term: P(small or large) = 0.45 + 0.2 = 0.65. Adding straight is right here. **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. That finds a missing one: large is 1 - 0.45 - 0.35 = 0.2. The same idea gives an event not happening: 1 minus the chance that it does. So ask first: could both happen to one prize? A toy can also be red, so those overlap. A prize cannot be small and large, so those do not. Independent is different: one happening does not change the chance of the other. A coin toss and a prize draw are independent and can happen together. When both have some chance, exclusive events cannot be independent, since one rules out the other. Covering every outcome is separate from sharing none: above 5 and below 9 cover cards yet share 6, 7 and 8. Adding without subtracting counts the overlap twice. Subtracting when there is none throws away prizes counted only once. **Recap.** P(A or B) = P(A) + P(B) - P(A and B), where the subtraction undoes the double count of anything sitting in both. Two events that cannot both happen are mutually exclusive, there is nothing in both, and the last term is zero. Independent is different: one happening does not change the chance of the other. When both have some chance, exclusive events cannot be independent, since one rules out the other. A complete set of outcomes that cannot overlap has probabilities adding to 1, which finds a missing one and gives the chance of an event not happening. Always ask first whether both could happen at once. ## 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)