Final answer:
The question involves determining whether events A and B are independent in probability, which is the case here because P(A AND B) equals the product of P(A) and P(B).
Step-by-step explanation:
The question is asking whether two events, A and B, are independent based on their individual probabilities and the probability of their intersection. If events A and B are independent, then the following should be true: P(A AND B) = P(A) × P(B). In this case, we are given P(A) = 0.4 and P(B) = 0.2, and thus their product is 0.08. If P(A AND B) is also 0.08, the events are independent since the product of their individual probabilities equals the probability of their intersection.
However, the condition for events to be mutually exclusive is different. If A and B are mutually exclusive, then P(A AND B) = 0. Therefore, if A and B are both mutually exclusive and independent, it implies that one of the events has a probability of 0, which is not the case here. Hence, for events A (learning Spanish) and B (learning German), they could be independent, but not mutually exclusive, given the probabilities provided.No, the statement is false. For two languages A and B, the concatenation of (AUB)BA is not equal to AB(BUA)*. Let's consider an example: A = {0,1} and B = {1,2}. (AUB)BA = {0,1,2,1}BA = {0,1,2,1}, and AB(BUA)* = {0,1}B{1,2}*(2,1)* = {0,1,2,1}* = {0,1,2,1,2,1,2,1,2,...}. These two languages are not equal, so the statement is false.