Testing for Independence

Probability · Axiom Academy

EXAMPLE Testing for Independence Using the independence criterion P(A ∩ B) = P(A) · P(B) Events A and B with their joint probabilities: Excellent work! You've completed this example. Here's what we learned: Independence definition: Events A and B are independent if and only if P(A ∩ B) = P(A) · P(B). If this equality holds, knowing one event occurred doesn't change the probability of the other. How to test: (1) Find the marginal probabilities P(A) and P(B) from row and column totals, (2) Find P(A ∩ B) from the joint cell, (3) Check if P(A ∩ B) = P(A) · P(B). These events are independent: P(A ∩ B) = 0.2 equals P(A) · P(B) = 0.5 × 0.4 = 0.2, so knowing B occurred doesn't change the probability of A. Alternative interpretation: Independence also means P(A|B) = P(A). You could verify: P(A|B) = P(A ∩ B) / P(B) = 0.2 / 0.4 = 0.5 = P(A). ✓ Symmetric property: If A is independent of B, then B is independent of A. Also, A c is independent of B, and A is independent of B c . Testing for independence is crucial in probability and statistics. Practice with different probability tables to build intuition!

This is the written version of the interactive lesson above. See the full Probability course.