If A and B are independent with P(A)=0.5 and P(B)=0.4, what is P(A∩B)?

If A and B are independent with P(A)=0.5 and P(B)=0.4, what is P(A∩B)?

• A. 0.2
• B. 0.5
• C. 0.4
• D. 0.8

Answer: (A)

Think of independence as a rule that the chance of both events happening equals the product of their separate chances. Since A and B do not affect each other, P(A∩B) = P(A)P(B) = 0.5 × 0.4 = 0.2. This value is the probability that both occur and it cannot exceed either individual probability, so it must be at most 0.4. The other numbers correspond to probabilities of single events or to a union, not the intersection. For example, 0.5 and 0.4 are P(A) and P(B) themselves, not the joint probability; and 0.8 would violate the fact that the intersection cannot be larger than the smaller probability. If you considered the union, you’d get P(A∪B) = 0.5 + 0.4 − 0.2 = 0.7, which is not among the choices.

Think of independence as a rule that the chance of both events happening equals the product of their separate chances. Since A and B do not affect each other, P(A∩B) = P(A)P(B) = 0.5 × 0.4 = 0.2. This value is the probability that both occur and it cannot exceed either individual probability, so it must be at most 0.4.

The other numbers correspond to probabilities of single events or to a union, not the intersection. For example, 0.5 and 0.4 are P(A) and P(B) themselves, not the joint probability; and 0.8 would violate the fact that the intersection cannot be larger than the smaller probability. If you considered the union, you’d get P(A∪B) = 0.5 + 0.4 − 0.2 = 0.7, which is not among the choices.