Back to Understand independence via the product rule — Problem 5 · Task Set 19

Exercises: Understand Independence Through the Product Rule

Work through each section in order. For independence tests, compute $P(A)$, $P(B)$, and $P(A \text{ and } B)$, then compare $P(A \text{ and } B)$ to the product $P(A) \cdot P(B)$. For explanation problems, write in complete sentences.

Grade 10·21 problems·~35 min·Common Core Math - HS Statistics and Probability·group·hss-cp-a-2

Work through problems with immediate feedback

Fluency Practice

Test each pair of events using the product rule.

You are told $P(A) = 0.5$ , $P(B) = 0.4$ , and $P(A \text{ and } B) = 0.25$ . Are $A$ and $B$ independent?

No — $P(A) \cdot P(B) = 0.20 \neq 0.25$ , so the product rule fails and the events are dependent.

Yes — both probabilities are positive, so the events must be independent.

Yes — $0.25$ is close enough to $0.20$ that we call them independent.

Cannot decide without knowing the experiment.