--- 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: 2026-11-12 --- > **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`. # 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 Given B, restrict the sample space to outcomes in B. When P(B) > 0, P(A given B) = P(A and B) / P(B). The numerator is the joint chance that both happen; the denominator is the chance of B. With joint chance 0.15 and P(B) = 0.5, the conditional chance is 0.15 / 0.5 = 0.3. This ratio is undefined when P(B) = 0. Along a tree path, multiply the first chance by the conditional chance of the second to find the joint chance. **Example.** In a high-risk group, 10 percent carry the condition: the base rate is 0.1. The test flags 90 percent of carriers: sensitivity is 0.9. It clears 90 percent of healthy people, so the false-positive rate, the chance it flags a healthy person, is 0.1. True positives: 0.1 × 0.9 = 0.09. False positives: 0.9 × 0.1 = 0.09. Total positives: 0.18. Given a positive, the carrier chance is 0.09 / 0.18 = 0.5. Divide true positives by all positives. *(drawing: Picture 20 positive results from this group: about 10 come from people who carry the condition and 10 from people who do not.)* Independence means P(A and B) = P(A) × P(B). When P(B) > 0, this also means P(A given B) = P(A): learning B changes nothing about A's chance. The product definition still works when P(B) = 0. Mutual exclusivity means P(A and B) = 0. Events with positive chances cannot be both independent and mutually exclusive, since 0 cannot equal a positive product. With at least one probability 0, a mutually exclusive pair is also independent. **Example.** Split the chance of being late by how you travel: P(bus) 0.4, P(late given bus) 0.25, P(late given car) 0.1. The bus branch gives 0.4 times 0.25, which is 0.1. The car branch gives 0.6 times 0.1, which is 0.06. Add the branches: 0.1 plus 0.06 is 0.16. A positive result depends on the base rate, sensitivity and false-positive rate. Multiply the carrier proportion by sensitivity to find the true-positive proportion. Multiply the healthy proportion by the false-positive rate to find the false-positive proportion. When a condition is rare, the larger healthy group can produce more false positives than true ones. This is not automatic: a sufficiently small false-positive rate can make most positives true. **Recap.** Conditioning narrows the world to what you know. Independence means learning one event changes nothing about the other. Bayes' theorem reverses which event is conditioned on. *Original lesson, flashcard and question revisions authored for this dataset.* ## 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) ## Optional connections - [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)