Pair wise independence does not imply mutual independence. Assume that the particular scales used in dry-batching a concrete mix are such that both aggregate and cement weights are subject to error. The two weights are measured independently. The aggregate is equally likely to be measured exactly correct or 20 pounds too large. The cement is equally likely to be measured exactly correct or 20 pounds too small. The total weight of aggregate and cement is desired. Let event A be no error in aggregate weight measurement, event B be no error in cement weight, and event C be no error in total weight.
(a) Show that the pairs of events A and B, A and C, and B and C are independent.
(b) Use (a) as a counterexample to show that pairwise independence does not imply mutual independence.
(c) Is the inverse statement true?