--- title: "Independent Events and the Multiplication Rule" description: "Use P(A and B) = P(A) × P(B) for independent events, and check whether learning one event happened changes the other's probability." canonical: https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN source: https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN.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`. # Independent Events and the Multiplication Rule Use P(A and B) = P(A) × P(B) for independent events, and check whether learning one event happened changes the other's probability. Subject: Mathematics · Area: Probability · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN ## Ready when they can - Multiply probabilities for two independent events - Test independence by checking whether one outcome changes the other's probability - Explain when to multiply for 'and' and how overlap affects addition for 'or' ## Lesson: When two things both have to happen At the prize table, 'or' counted either description. Now two stalls ask for 'and': winning both. In this model the results are independent. P(win coconuts) = 0.2 and P(win tins) = 0.25. Winning both cannot be more likely than winning either game alone. Adding gives 0.45, which is above both chances, so addition cannot answer this 'and' question. *(drawing: In a 100-part chance model, 20 parts win the coconuts. Of those 20, the first group of 5 also wins the tins.)* In our chance model, 0.2 of 100 parts is 20. Independence means 0.25 of those 20 parts, or 5, also wins the tins. So P(both) = 5/100 = 0.05 = 0.2 × 0.25. The rule is P(A and B) = P(A) × P(B) for independent events. For three mutually independent events, multiply all three probabilities. Each independent fair coin flip has a 0.5 chance of heads. To find P(B), divide P(A and B) by P(A), but only when independence is given and P(A) > 0. Independent means learning one event happened leaves the other's probability unchanged. Different equipment alone does not prove this; a player's skill could link two games. With all three probabilities, test P(A and B) = P(A) × P(B). Mutually exclusive events with positive probabilities are dependent: learning one happened makes the other's chance 0. If either probability is 0, the product test holds and the events are independent. *(drawing: Test what knowing the first result tells you about the second.)* **Example.** A drum holds 20 tickets, 5 winners. Each draw picks uniformly at random. After a winning ticket is kept, 19 tickets remain with 4 winners: the next chance is 4/19, not 5/20. The draws are dependent. For two wins, multiply the first chance by the updated chance: (5/20) × (4/19), not 0.25 × 0.25. Replacing the ticket and mixing restores the original chance, making the draws independent. For independent events, 'and' means multiply. A probability is between 0 and 1, so multiplying by it cannot increase a value. The chance of both is at most either individual chance; 'or' gives at least either one. Equality is possible. For 'or', add and subtract overlap. Loss has chance 1 - P(win). Independent wins give independent losses. Multiply for neither; subtract that result from 1 for at least one. Earlier independent fair coin flips leave the next heads chance at 0.5. **Recap.** For independent events, P(A and B) = P(A) × P(B). Check whether learning one result changes the other's chance. In our raffle without replacement, update the second winning chance. ## Practice 33 questions on this page, each with its working shown. ## Needs first - [The Addition Rule and Mutually Exclusive Events](https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE) ## Opens up - [Tree Diagrams Without Replacement](https://lightmysky.com/learn/mathematics/tree-diagrams-without-replacement-mt_NfiAY_wMwZ) - [Conditional Probability, Independence and Bayes' Theorem](https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK) - [Discrete Random Variables and Probability Distributions](https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3)