Two production plants, A and B, make wire cables that are sent to a common distributor. 40% of the cables sent to the distributor come from Plant A, and the remaining 60% from Plant B. Among the cables produced at Plant A, 95% meet the strength specifications; among the cables produced at Plant B, 98% meet the strength specification.
a. The distributor selects one cable at random from among all cables in stock. If the cable selected is found to meet the strength specifications, what is the probability that the cable was produced at Plant A?
b. Let A be the event that a cable is manufactured at Plant A; similarly, B be the event that a cable is manufactured at Plant B. Let Y be the event that a cable meets the strength specifications; and let N be the event that a cable does not the strength specifications. (i) Are events A and Y independent? Give reasons; (ii) Are events B and N independent? Give reasons.