The product rule is used to solve problems in which

Question

BlackTom AI · Accepted Answer

The product rule applies when events are independent.

1. Relevant knowledge points and concepts
- In probability, independence means the occurrence of one event does not change the probability of the other. For independent events A and B: P(A and B) = P(A) × P(B).
- In counting (combinatorics), the product rule states that if one task can be done in m ways and an independent second task can be done in n ways, the combined task can be done in m × n ways. Independence here means the number of choices for one task does not depend on the choice for the other.

2. Complete reasoning process step by step
- Identify whether events or choices affect each other. If one outcome does not change the possibilities or probabilities of the other, they are independent.
- Apply multiplication: multiply probabilities for probability problems or multiply counts for counting problems.

3. Calculations with intermediate steps (examples)
- Probability example: Toss a fair coin (P(heads) = 1/2) and roll a fair six-sided die (P(4) = 1/6). These are independent.
  Step 1: P(heads) = 1/2.
  Step 2: P(4 on die) = 1/6.
  Step 3: P(heads AND 4) = (1/2) × (1/6) = 1/12.
- Counting example: 3 shirt choices and 4 pant choices, independent choices.
  Number of outfits = 3 × 4 = 12.

4. Why the correct answer is correct
- The product rule explicitly requires independence so that probabilities or counts multiply without conditional adjustment. If events influence each other, probabilities must be multiplied by conditional probabilities (P(A and B) = P(A) × P(B|A)), not by the simple product rule for independent events.

5. Why the other options are incorrect
- "Events are mutually exclusive.": Mutually exclusive events cannot occur together, so "and" probability is zero; product rule for joint occurrence is not applicable.
- "Events can occur in an unspecified order.": Order being unspecified is unrelated to whether you multiply probabilities or counts; independence is the key condition.
- "Events are not independent.": If not independent, you must use conditional probabilities (P(A and B) = P(A) × P(B|A)), so the simple product rule does not apply.

6. Conclusion
Therefore, the product rule is used when events are independent, which matches the provided correct answer: "events are independent."

The product rule is used to solve problems in which单项选择题

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