Let P(A) = 0.2, P(B) = 0.4, and P(A and B) = 0.08. Are events A and B independent?

Question

Let P(A) = 0.2, P(B) = 0.4, and P(A and B) = 0.08.  Are events A and B independent?

BlackTom AI · Accepted Answer

Check whether P(A ∩ B) equals P(A)P(B). Independence of two events A and B is defined by the equality P(A ∩ B) = P(A)P(B). An equivalent condition is P(A | B) = P(A) (or P(B | A) = P(B)) when the conditioning probability is nonzero.

1. Relevant knowledge points and concepts
- Definition of independence: A and B are independent iff P(A ∩ B) = P(A)P(B).
- Conditional probability relation: P(A | B) = P(A ∩ B) / P(B); if this equals P(A) then A and B are independent.

2. Step-by-step reasoning and calculations
- Given: P(A) = 0.2, P(B) = 0.4, P(A ∩ B) = 0.08.
- Compute P(A)P(B): 0.2 × 0.4 = 0.08.
- Compare: P(A ∩ B) (given) = 0.08 and P(A)P(B) (computed) = 0.08. They are equal.

3. Intermediate steps shown
- Multiplication: 0.2 × 0.4 = (2/10) × (4/10) = 8/100 = 0.08.
- Conditional check (optional confirmation): P(A | B) = P(A ∩ B)/P(B) = 0.08 / 0.4 = 0.2 = P(A). Similarly P(B | A) = 0.08 / 0.2 = 0.4 = P(B).

4. Why the correct answer is correct
Because the defining equality for independence holds exactly: P(A ∩ B) = P(A)P(B) = 0.08, so A and B are independent.

5. Why the other options are incorrect
- "No, because P(A and B) isn't 0.": Independence does not require the intersection to be 0; many independent events have nonzero intersection.
- "No, because P(A and B) is not equal to P(A) + P(B).": That comparison is irrelevant; P(A ∪ B) relates to P(A) + P(B) − P(A ∩ B), and independence is not about sums.
- "We don't have enough information to know.": We do have sufficient information: the three probabilities allow a direct check of the independence condition.

6. Conclusion
Since 0.08 = 0.2 × 0.4, A and B are independent. This matches the provided correct answer: "Yes, because P(A and B) = P(A)P(B)."

Let P(A) = 0.2, P(B) = 0.4, and P(A and B) = 0.08. Are events A and B independent?单项选择题

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