Back to Understand independence via the product rule — Problem 3 · 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.

Using the same die events, $A = $ "even" and $B = $ "greater than 2," we have $P(A) = \frac{1}{2}$ , $P(B) = \frac{2}{3}$ , and $P(A \text{ and } B) = \frac{1}{3}$ . Are $A$ and $B$ independent?

Yes — $P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3} = P(A \text{ and } B)$ , so the product rule holds.

No — the two events come from the same die, so they cannot be independent.

No — $P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{2}{3} = \frac{2}{3} \neq \frac{1}{3}$ .

Cannot tell without listing the full sample space again.