If two events A and B are independent, what is P(A ∩ B)?

If two events A and B are independent, what is P(A ∩ B)?

• A. P(A) + P(B)
• B. P(A) × P(B)
• C. P(B) − P(A)
• D. 1

Answer: (B)

Independence means the occurrence of one event does not affect the probability of the other. For two independent events, the probability that both occur is the product of their individual probabilities: P(A ∩ B) = P(A) × P(B|A) = P(A) × P(B), since P(B|A) = P(B) when A and B are independent. So if P(A) = 0.6 and P(B) = 0.5, the probability both happen is 0.6 × 0.5 = 0.30. This is why the product is the correct expression. The sum P(A) + P(B) would apply only if the events were mutually exclusive (they cannot happen together), in which case P(A ∩ B) would be 0, not the sum. The other options don’t generally describe P(A ∩ B) for independent events, and 1 would only occur if both events were certain to happen.

Independence means the occurrence of one event does not affect the probability of the other. For two independent events, the probability that both occur is the product of their individual probabilities: P(A ∩ B) = P(A) × P(B|A) = P(A) × P(B), since P(B|A) = P(B) when A and B are independent.

So if P(A) = 0.6 and P(B) = 0.5, the probability both happen is 0.6 × 0.5 = 0.30. This is why the product is the correct expression.

The sum P(A) + P(B) would apply only if the events were mutually exclusive (they cannot happen together), in which case P(A ∩ B) would be 0, not the sum. The other options don’t generally describe P(A ∩ B) for independent events, and 1 would only occur if both events were certain to happen.