Back to Exercise: Understand independence via the product rule

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

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Warm-Up: Probability Foundations

These problems review skills you already know.

A fair coin is flipped twice. Listing the sample space $\{HH, HT, TH, TT\}$ , what is $P(\text{both heads})$ ? Express your answer as a fraction.

Which statement about the word "independent" in probability is correct?

Independence is a relationship between two events: $A$ is independent of $B$ .

Independence is a property a single event has on its own.

An event is independent whenever its probability is greater than $0.5$ .

Independence only describes events from two separate experiments.

Two events $A$ and $B$ are mutually exclusive when they cannot both occur. What is $P(A \text{ and } B)$ for mutually exclusive events?

$P(A \text{ and } B) = 0$

$P(A \text{ and } B) = P(A) \cdot P(B)$

$P(A \text{ and } B) = P(A) + P(B)$

$P(A \text{ and } B) = 1$

Fluency Practice

Test each pair of events using the product rule.

Two fair coins are flipped. Let $A = $ "first coin is heads" and $B = $ "second coin is heads." Compute $P(A) \cdot P(B)$ . Express your answer as a fraction.

A fair die is rolled once. Let $A = $ "even" $= \{2, 4, 6\}$ and $B = $ "greater than 2" $= \{3, 4, 5, 6\}$ . Compute $P(A \text{ and } B)$ directly by counting the outcomes in both events. Express your answer as a fraction.

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.

A spinner lands on red with probability $0.4$ . It is spun twice, and the two spins are independent. What is the probability it lands on red both times? Give a decimal.

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.

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