For each of the following statements about events A,B, and C defined on a common sample space, determine whether it is true or false.
1. Suppose that A,B , and C are pairwise independent. Then, A∩C is independent of B .
2. Suppose that A,B, and C are pairwise independent. Then, A,B, and C are independent.
1. Well, let's think about this. If A, B, and C are pairwise independent, it means that any pair of the events A and B, A and C, or B and C are independent of each other. However, this doesn't necessarily mean that the intersection of A and C, A∩C, is independent of B. So, this statement is FALSE. Sorry, A∩C isn't so independent after all!
2. Ah, I see what you did there. If A, B, and C are pairwise independent, it means that any pair of the events A and B, A and C, or B and C are independent of each other. If this holds true for all three pairs, then A, B, and C are indeed independent events. So, this statement is TRUE. Congratulations, A, B, and C are truly independent! They're like three peas in a pod, living their lives without any interference.
To determine the truth or falsehood of the statements, let's analyze each one:
1. Suppose that A, B, and C are pairwise independent. Then, A∩C is independent of B.
This statement is TRUE.
Pairwise independence means that the probability of one event occurring does not affect the probability of another event occurring. In this case, it means that the probabilities of events A and B, A and C, and B and C occurring are independent of each other.
So, if A, B, and C are pairwise independent, it follows that A∩C is independent of B. The intersection of A and C, i.e., A∩C, does not have any impact on the probability of event B occurring. Therefore, A∩C is independent of B.
2. Suppose that A, B, and C are pairwise independent. Then, A, B, and C are independent.
This statement is FALSE.
Pairwise independence alone is not sufficient to imply independence among all three events. It only guarantees that pairs of events are independent, but it does not necessarily imply independence among all three events together.
To determine independence among all three events (A, B, and C), you would need to consider the joint probabilities of the events occurring together. If the joint probabilities satisfy the definition of independence, then the events are independent. But the given information of pairwise independence does not provide enough information to conclude independence among all three events.
To determine whether the statements are true or false, we need to understand the definitions of pairwise independence and independence, and then apply them to the given statements.
1. Pairwise Independence: Events A, B, and C are said to be pairwise independent if any two of them, when taken together, are independent. In other words, knowing the occurrence or non-occurrence of one event gives no information about the occurrence or non-occurrence of the other event(s).
To determine if statement 1 is true or false, we need to check if A∩C is independent of B, given that A, B, and C are pairwise independent.
If A, B, and C are pairwise independent, then A and B are independent, A and C are independent, and B and C are independent. However, this does not guarantee that A∩C is independent of B.
To check if A∩C is independent of B, we need to calculate the conditional probability P(A∩C | B) and compare it with the marginal probability P(A∩C).
If P(A∩C | B) = P(A∩C), then A∩C is independent of B. If P(A∩C | B) ≠ P(A∩C), then A∩C is not independent of B.
2. Independence: Events A, B, and C are said to be independent if the occurrence or non-occurrence of any one of the events gives no information about the occurrence or non-occurrence of the other events. In other words, knowing the occurrence or non-occurrence of any event does not affect the probabilities of the other events.
To determine if statement 2 is true or false, we need to check if A, B, and C are independent, given that they are pairwise independent.
If A, B, and C are pairwise independent, then A and B are independent, A and C are independent, and B and C are independent. However, this does not guarantee that A, B, and C are independent.
To check if A, B, and C are independent, we need to calculate the joint probabilities P(A∩B∩C), P(A)P(B), P(A)P(C), and P(B)P(C) and compare them.
If P(A∩B∩C) = P(A)P(B)P(C), then A, B, and C are independent. If P(A∩B∩C) ≠ P(A)P(B)P(C), then A, B, and C are not independent.
In summary, to determine whether the given statements are true or false, you need to check the conditional probabilities and joint probabilities as explained above.