--- title: "Conditional Probability, Independence and Bayes' Theorem" description: "Conditioning restricts the sample space. Independence is the case where it changes nothing, and Bayes' theorem reverses the conditioning, which is how evidence updates a belief." canonical: https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK source: https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK.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`. # Conditional Probability, Independence and Bayes' Theorem Conditioning restricts the sample space. Independence is the case where it changes nothing, and Bayes' theorem reverses the conditioning, which is how evidence updates a belief. Subject: Mathematics · Area: Probability · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK ## Ready when they can - Compute a conditional probability from a joint and a marginal - Apply Bayes' theorem to a screening test and interpret the result - Distinguish independent events from mutually exclusive ones ## Lesson: Conditioning, independence and Bayes You narrow the world when you condition. P(A given B) equals P(A and B) divided by P(B). With joint 0.24 and P(B) 0.6, the conditional chance is 0.24 divided by 0.6, which is 0.4. Along a tree path you multiply: the chance of both equals the first chance times the conditional chance of the second. **Example.** Take a high-risk group where 10 percent carry a condition. A test catches 90 percent of cases and clears 90 percent of healthy people. True positives are 0.1 times 0.9, which is 0.09. False positives are 0.9 times 0.1, which is also 0.09. Total positives are 0.18, so Bayes gives 0.09 divided by 0.18, which is 0.5. Half of all positives are real cases. Independence means learning changes nothing: P(A and B) equals P(A) times P(B), so P(A given B) equals P(A). Mutual exclusivity means both cannot happen, so P(A and B) equals 0. With positive chances a pair cannot be both, since 0 cannot equal a positive product. **Example.** Split lateness over weather: P(rain) 0.3, P(late given rain) 0.5, P(late given no rain) 0.2. The rain branch gives 0.3 times 0.5, which is 0.15. The dry branch gives 0.7 times 0.2, which is 0.14. Add the branches: 0.15 plus 0.14 is 0.29. **Tip.** Read a positive test against the base rate. When the condition is rare, healthy people outnumber cases, so false positives can match true ones. That is why an accurate test for a rare condition still returns many false alarms. **Recap.** Conditioning narrows the world, independence ignores the news, and Bayes reverses the question. ## Practice 13 questions on this page, each with its working shown. ## Needs first - [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) - [Conditional Probability from Tables and Venn Diagrams](https://lightmysky.com/learn/mathematics/conditional-probability-from-tables-and-venn-diagrams-mt_Vx88ugViYP) ## Opens up - [Markov Chains and the Memoryless Assumption](https://lightmysky.com/learn/mathematics/markov-chains-and-the-memoryless-assumption-mt_2wDa_JxJ_o) - [Discrete Models Beyond the Binomial](https://lightmysky.com/learn/mathematics/discrete-models-beyond-the-binomial-mt_9zc5xYjDg_) - [Bayes' Rule and Updating a Belief](https://lightmysky.com/learn/computing/bayes-rule-and-updating-a-belief-mt_A4p5--f7Io) - [Chi-Square Tests for Categorical Data](https://lightmysky.com/learn/mathematics/chi-square-tests-for-categorical-data-mt_yxc87pzaXX)