Unit 12 · Lesson 03 · Open-Age
Conditional Probability, Independence & Bayes Reasoning
Conditional probability updates the sample space after information is known. Independence means P(A|B)=P(A), not merely that events are different.
Learning objectives
- Explain the central structure of conditional probability, independence & bayes reasoning using precise mathematical language.
- Apply the governing relationship and verify results with an alternate representation or substitution.
- State domain, unit, precision, and context restrictions whenever they affect the result.
- Analyze a misconception and communicate a correction with evidence.
Essential question: How do counting, probability models, distributions, and sampling support responsible conclusions under uncertainty?
Governing relationship
P(A|B)=P(A∩B)/P(B).
Conditional probability updates the sample space after information is known. Independence means P(A|B)=P(A), not merely that events are different.
Worked examples
Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.
Guided and independent practice
- Compute conditional probability from a table.
- Test independence.
- Use a probability tree.
- Create one original example involving conditional probability, independence & bayes reasoning and solve it completely.
- Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
- Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.
Differentiated pathways
Foundation
Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.
Core
Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.
Extended
Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.
Application laboratory
Analyze screening, reliability, or category data without confusing base rates.
Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.
Misconception and correction
Common misconception: Reversing P(A|B) and P(B|A).
Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.
Mastery evidence
- Accurate symbolic work with preserved equivalence.
- Correct domain, interval, unit, and precision statements.
- At least one independent verification.
- A correction explaining why an incorrect approach fails.
- A clear application conclusion that distinguishes mathematical result from recommendation.
Reflection
What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?