Lesson 3: 5.3 Conditional Probability and Independence

1. How does knowing that one event has occurred change the "denominator" of our probability?

2. What is the mathematical difference between P(A) and P(A|B)?

UNC-2.A (General Multiplication Rule), CPV-1.A (Tree Diagrams), UNC-2.A.4 (Independence test), HSS-CP.A.2, HSS-CP.A.3, HSS-CP.A.5.

Description
This section covers the probability of an event given that another has occurred. It introduces the formula $P(A

Purpose
To equip students with the tools to handle "dependent" scenarios (like sampling without replacement). This logic is the precursor to P-values—the probability of getting our data given that the null hypothesis is true.

DOK Level
Level 3 (Strategic Thinking): Students must select the appropriate tool (Tree Diagram vs. Formula) for complex word problems and provide formal mathematical justifications for whether events are independent.

Struggling Learners: Focus on the "Restricted Sample Space" approach. Instead of using the formula immediately, have them physically circle the row or column in a two-way table that represents the "given" condition. This makes it clear that the total (denominator) has shrunk.

Advanced Learners: Have them solve a "false positive" medical testing problem (the classic Bayes' Theorem application) using a tree diagram to see how a rare disease can have a low predictive value even with a "99% accurate" test.

ELL Learners: Use Tree Diagrams as a primary visual scaffold. The branching structure provides a non-linguistic way to show "what happens next," and the word "Given" can be visually associated with the specific branch they are currently standing on.

Mastery Based assessments