The probability of Event A occurring is 0.3 and the probability of Event B occurring is 0.5. If Event A and Event B are independent, then what is the joint probability of Event A and Event B both occurring?

Question

BlackTom AI · Accepted Answer

For independent events, P(A and B) = P(A) × P(B), so compute the product of the two probabilities.

1) Relevant knowledge points and concepts
- Independence: Two events A and B are independent if the occurrence of one does not affect the probability of the other. For independent events the multiplication rule applies: P(A ∩ B) = P(A) · P(B).
- Probabilities must lie between 0 and 1; the joint probability cannot exceed either marginal probability (it is ≤ min{P(A), P(B)} in general).

2) Step-by-step reasoning
- Given P(A) = 0.3 and P(B) = 0.5 and that A and B are independent.
- Apply the multiplication rule for independent events.

3) Calculations with intermediate steps
- Convert to fractions optionally: P(A) = 0.3 = 3/10, P(B) = 0.5 = 1/2.
- Multiply: P(A ∩ B) = (3/10) × (1/2) = 3/20.
- Convert back to decimal: 3/20 = 0.15.

4) Why the correct answer is correct
- Independence justifies using the product rule; the arithmetic yields 0.15. This value is ≤ min(0.3, 0.5) = 0.3, so it is a valid joint probability and consistent with the definition of independence.

5) Why other common incorrect answers would be wrong
- Adding probabilities (0.3 + 0.5 = 0.8) would give P(A ∪ B) only if A and B were mutually exclusive and even then must subtract any overlap; it is not the joint probability.
- Subtracting (0.5 − 0.3 = 0.2) or taking the larger marginal are not supported by probability rules for the intersection of independent events.
- Any number greater than 0.3 would violate the fact that P(A ∩ B) ≤ P(A).

6) Conclusion
- Therefore the joint probability P(A and B) = 0.15, which matches the provided answer 0.15.

The probability of Event A occurring is 0.3 and the probability of Event B occurring is 0.5. If Event A and Event B are independent, then what is the joint probability of Event A and Event B both occurring?单项选择题

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