Final answer:
In summary, without making any assumptions about the dependence between events A and B, the probability P(AB) must be within the range of 0.1 and 0.4, due to the constraints provided by the addition rule (minimum bound) and the nature of probabilities (maximum bound).
Step-by-step explanation:
To find the range of possible values for P(AB), the probability of events A and B occurring together, we need to consider two scenarios: when the events are independent and when they are not independent.
In the case of independent events, the probability of both A and B occurring is the product of their individual probabilities. Therefore, we can calculate P(AB) when A and B are independent by multiplying: P(A) × P(B) = 0.4 × 0.7 = 0.28. Since we make no further assumptions on A and B, this result serves only as a reference point.
If A and B are not independent, this means the occurrence of one event affects the probability of the occurrence of the other. In the most extreme case where one event always implies the other (perfect dependence), the probability of both occurring is the minimum of the two probabilities. Hence, the lower bound for P(AB) is max(0, P(A) + P(B) - 1) = max(0, 0.4 + 0.7 - 1) = 0.1. Conversely, the upper bound is just the smaller of the two probabilities, i.e., P(AB) ≤ min(P(A), P(B)) = min(0.4, 0.7) = 0.4.
Combining these constraints, we conclude that 0.1 ≤ P(AB) ≤ 0.4 regardless of the dependency between A and B.