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.

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To determine the truth value of these statements, we need to understand the definitions of pairwise independence and independence.

1. Pairwise independence: Three events A, B, and C are said to be pairwise independent if any two events are independent of each other. In other words, for any pairs of events (A, B), (A, C), and (B, C), the probability of their intersection is equal to the product of their individual probabilities:

P(A∩B) = P(A) * P(B)
P(A∩C) = P(A) * P(C)
P(B∩C) = P(B) * P(C)

Now let's determine whether A∩C is independent of B. To show that they are independent, we need to verify the following equality:

P((A∩C)∩B) = P(A∩C) * P(B)

To find the left-hand side probability, we can rewrite it as:

P((A∩C)∩B) = P((A∩C)∩B) / P(B) * P(B)

Now, if A∩C is independent of B, we should have:

P((A∩C)∩B) / P(B) = P(A∩C) * P(B) / P(B)

If the above equality holds true, then the statement is true. Otherwise, it is false.

2. For this statement, we need to verify if A, B, and C are mutually independent events. For three events to be independent, the joint probability of the intersection of all three events should be equal to the product of each event's individual probabilities:

P(A∩B∩C) = P(A) * P(B) * P(C)

If the above equality holds true, then the statement is true. Otherwise, it is false.

By calculating the necessary probabilities and comparing them to the definitions of independence, you can determine the truth or falsity of these statements.